This section has 4 verses.

Titus 2:11-14
11 For the grace of God that bringeth salvation hath appeared to all men,
12 Teaching us that, denying ungodliness and worldly lusts, we should live soberly, righteously, and godly, in this present world;
13 Looking for that blessed hope, and the glorious appearing of the great God and our Saviour Jesus Christ;
14 Who gave himself for us, that he might redeem us from all iniquity, and purify unto himself a peculiar people, zealous of good works.

We will begin with the Barclay commentary.

THE MORAL POWER OF THE INCARNATION

First the verses paraphrased:

Titus 2:11–14  
For the grace of God, which brings salvation to all men, has appeared, schooling us to renounce godlessness and worldly desires for forbidden things, and to live in this world prudently, justly and reverently, because we expectantly await the realization of our blessed hope – I mean the glorious appearing of our great God and Saviour Jesus Christ, who gave himself for us to redeem us from the power of all lawlessness, and to purify us as a special people for himself, a people eager for all fine works.

THERE are few passages in the New Testament which so vividly set out the moral power of the incarnation as this does. Its whole stress is the miracle of moral change which Jesus Christ can work.

This miracle is repeatedly expressed here in the most interesting and significant way. Isaiah once exhorted his people: ‘Cease to do evil, learn to do good’ (Isaiah 1:16–17). First, there is the negative side of goodness, the giving up of that which is evil and the liberation from that which is low; second, there is its positive side, the acquisition of the shining virtues which mark the Christian life.

First, there is the renunciation of all godlessness and worldly desires. What did Paul mean by worldly desires? The early Church [member], John Chrysostom, said that worldly things are things which do not pass over with us into [the Kingdom] but are dissolved together with this present world. People are very short-sighted if they set their hearts and expend all their energies on things which they must leave behind when they quit this world. But an even simpler interpretation of worldly desires is that they are for things we could not show to God. It is only Christ who can make not only our outward life but also our inward heart fit for God to see.

That was the negative side of the moral power of the incarnation; now comes the positive side. Jesus Christ makes us able to live with the prudence which has everything under perfect control, and which allows no passion or desire more than its proper place; with the justice which enables us to give both to God and to our neighbours that which is their due; with the reverence which makes us live in the awareness that this world is nothing other than the temple of God.

The dynamic of this new life is the expectation of the coming of Jesus Christ. When a royal visit is expected, everything is cleaned and decorated and made fit for royal eyes to see. Christians are men and women who are always prepared for the coming of the King of Kings.

Finally, Paul goes on to sum up what Jesus Christ has done, and once again he does it first negatively and then positively.

Jesus has redeemed us from the power of lawlessness, that power which makes us sin.

Jesus can purify us until we are fit to be the special people of God. The word we have translated as special (periousios) is interesting. It means reserved for, and it was used for that part of the spoils of a battle or a campaign which the king who had conquered set apart especially for himself. Through the work of Jesus Christ, Christians become fit to be the special possessions of God.

The moral power of the incarnation is a tremendous thought. Christ not only liberated us from the penalty of past sin; he can enable us to live the perfect life within this world of space and time; and he can so cleanse us that we become fit in the life to come to be the special possession of God. ~Barclay Commentary

Now to the other commentaries. We begin with the general and go to the specific.

This from the Matthew Henry Main. This commentary covers verses 11-14. I will give you the first part of this commentary which discusses the meaning of verse 11.

Here we have the grounds or considerations upon which all the foregoing directions are urged, taken from the nature and design of the gospel, and the end of Christ's death.

I. From the nature and design of the gospel. Let young and old, men and women, masters and servants, and Titus himself, let all sorts do their respective duties, for this is the very aim and business of Christianity, to instruct, and help, and form persons, under all distinctions and relations, to a right frame and conduct. For this,

1. They are put under the dispensation of the grace of God, so the gospel is called, Ephesians 3:2. It is grace in respect of the spring of it - the free favour and good-will of God, not any merit or desert in the creature; as manifesting and declaring this good-will in an eminent and signal manner; and as it is the means of conveying and working grace in the hearts of believers. Now grace is obliging and constraining to goodness: Let not sin reign, but yield yourselves unto God; for you are not under the law, but under grace, Romans 6:12-14. The love of Christ constrains us not to live to self, but to him (2 Corinthians 5:14-15); without this effect, grace is received in vain.

Quoted verses:
Ephesians 3:2
If ye have heard of the dispensation [stewardship, administration] of the grace of God which is given me to you-ward:

Romans 6:12-14
12 Let not sin therefore reign in your mortal body, that ye should obey it in the lusts thereof.
13 Neither yield ye your members as instruments of unrighteousness unto sin: but yield yourselves unto God, as those that are alive from the dead, and your members as instruments of righteousness unto God.
14 For sin shall not have dominion over you: for ye are not under the law, but under grace.

2 Corinthians 5:14-15
14 For the love of Christ constraineth us; because we thus judge, that if one died for all, then were all dead:
15 And that he died for all, that they which live should not henceforth live unto themselves, but unto him which died for them, and rose again.

2. This gospel grace brings salvation (reveals and offers it to sinners and ensures it to believers) - salvation from sin and wrath, from death and hell [permanent death]. Hence it is called the word of life; it brings to faith, and so to life, the life of holiness now and of happiness hereafter. The law is the ministration of death, but the gospel the ministration of life and peace. This therefore must be received as salvation (its rules minded, its commands obeyed), that the end of it may be obtained, the salvation of the soul. And more inexcusable will the neglecters of this grace of God bringing salvation now be, since,

3. It hath appeared, or shone out more clearly and illustriously than ever before. The old dispensation was comparatively dark and shadowy; this is a clear and shining light; and, as it is now more bright, so more diffused and extensive also. For,

4. It hath appeared to all men; not to the Jews only, as the glory of God appeared at mount Sinai to that particular people, and out of the view of all others; but gospel grace is open to all, and all are invited to come and partake of the benefit of it, Gentiles as well as Jews. The publication of it is free and general: Disciple all nations: Preach the gospel to every creature. The pale [lack of brightness or brilliance] is broken down; there is no such enclosure now as formerly. The preaching of Jesus Christ, which was kept secret since the world began, now is made manifest, and by the scriptures of the prophets, according to the commandment of the everlasting God, made known to all nations for the obedience of faith, Romans 16:25-26. The doctrine of grace and salvation by the gospel is for all ranks and conditions of men (slaves and servants, as well as masters), therefore engaging and encouraging all to receive and believe it, and walk suitably to it, adorning it in all things. ~Matthew Henry Main

Quoted verse:
Romans 16:25-26
25 Now to him that is of power to stablish you according to my gospel, and the preaching of Jesus Christ, according to the revelation of the mystery, which was kept secret since the world began,
26 But now is made manifest, and by the scriptures of the prophets, according to the commandment of the everlasting God, made known to all nations for the obedience of faith:

Now the Matthew Henry Concise. This commentary covers the rest of Chapter 2, verses 11-15.

The doctrine of grace and salvation by the gospel, is for all ranks and conditions of men. It teaches to forsake sin; to have no more to do with it. An earthly, sensual conversation suits not a heavenly calling. It teaches to make conscience of that which is good. We must look to God in Christ, as the object of our hope and worship. A gospel conversation must be a godly conversation. See our duty in a very few words; denying ungodliness and worldly lusts, living soberly, righteously, and godly, notwithstanding all snares, temptations, corrupt examples, ill usage, and what remains of sin in the believer's heart, with all their hinderances. It teaches to look for the glories of another world. At, and in, the glorious appearing of Christ, the blessed hope of Christians will be complete: To bring us to holiness and happiness was the end of Christ's death. Jesus Christ, that great God and our Saviour, who saves not only as God, much less as Man alone; but as God-man, two natures in one person. He loved us, and gave himself for us; and what can we do less than love and give up ourselves to him! Redemption from sin and sanctification of the nature go together, and make a peculiar people unto God, free from guilt and condemnation, and purified by the Holy Spirit. All Scripture is profitable. Here is what will furnish for all parts of duty, and the right discharge of them. Let us inquire whether our whole dependence is placed upon that grace which saves the lost, pardons the guilty, and sanctifies the unclean. And the further we are removed from boasting of fancied good works, or trusting in them, so that we glory in Christ alone, the more zealous shall we be to abound in real good works. ~Matthew Henry Concise

Here are some things from the Biblical Illustrator. Each one is a recap of the discussion of the gospel.

The grace of God that bringeth salvation
The gospel

I. What is here said of its nature.

1. The name. “The grace of God.”
2. The subject “Bringing salvation.”
3. The manifestation. “Hath appeared.”

(1) None are excluded from its benefits.
(2) None are exempt from its appointments.

II. Its influence.
1. How the gospel teaches.
(1) Precept.
(2) Example.
(3) Motive.
(4) Real and spiritual operation and efficiency.

2. What the gospel teaches.
(1) What it teaches us to deny? Ungodliness and worldly lusts.
(2) What it teaches us to do? “To live soberly, righteously, and godly in the present world.”
(3) What it teaches us to expect? “Looking for that blessed hope and the glorious appearing of the great God and our Saviour Jesus Christ.”
(4) What it teaches us to acknowledge? “Who gave Himself,” etc. ~Biblical Illustrator

Here is another piece from the Biblical Illustrator about the salvation process we are discussing today:

The gospel of the grace of God

I. Its distinguishing characteristics. “The grace of God.”
1. The gift.
2. Its objects.
3. Its purpose.

II. The universality of its appearance.
1. Adapted for all.
2. Revealed for all.
3. To be proclaimed to all.

III. The inestimable boon which it bestows. “Salvation.”
1. From the condemning power of sin.
2. From the defilement of sin.
3. From the love of sin.
4. From the power of sin.
5. From the punishment of sin. ~Biblical Illustrator

Here is another recap of the discussion:

The grace of God in bringing salvation to all men

I. The origin of salvation.
1. Man did not deserve it.
2. It was unsolicited.
3. It was entirely the result of Divine grace.

The grace of God
(1) Made all the arrangements necessary for salvation. Devised the astounding plan. Fixed upon the means, time, etc. The grace of God
(2) Brought the author of salvation. “Ye know the grace of our Lord Jesus Christ,” etc. (2 Corinthians 8:9).
(3) It brought the message of salvation. Gospel is emphatically the gospel of the grace of God (Acts 20:24).
(4) It brings the application of salvation to the soul. We are called by His grace—justified freely by His grace—sanctified by His grace—kept and preserved by its grace—and the topstone is brought on amid ascriptions of Grace, grace unto it.”

Quoted verses:
2 Corinthians 8:9
For ye know the grace of our Lord Jesus Christ, that, though he was rich, yet for your sakes he became poor, that ye through his poverty might be rich.

Acts 20:24 ...Paul discussing that he was going to Jerusalem and not knowing the things that would befall him there
But none of these things move me, neither count I my life dear unto myself, so that I might finish my course with joy, and the ministry, which I have received of the Lord Jesus, to testify the gospel of the grace of God.

II. The extent of salvation. The grace of God bringeth salvation
1. To all classes and degrees of men. To the rich and the poor; noble and ignoble; monarch and the peasant; the ruler and the slave.
2. To men of all grades of moral guilt. It includes the moralist, and excludes not the profane.
3. To men of all ages.

III. The influence of salvation on the moral character of man. It teaches and enforces the necessity of
1. The abandonment of ungodliness and worldly lusts.
2. Sobriety of conduct.
3. Righteousness of life.
4. Godliness of heart.

Application:
1. How we should rejoice in the riches and fulness of Divine grace.
2. How necessary that we cordially receive the invaluable boon it presents.
3. And how important that we practically exemplify the moral lessons it communicates. ~Biblical Illustrator

Now to the specific commentaries.

Most commentaries break this down into two parts:

1] For the grace of God.
2] That bringeth salvation hath appeared to all men.

1] For the grace of God.

For the grace of God - The favor of God, shown to the undeserving; see the notes at Romans 1:7. ~Barnes Notes

Quoted verse:
Romans 1:7
To all that be in Rome, beloved of God, called to be saints: Grace to you and peace from God our Father, and the Lord Jesus Christ.

Let us read some of the commentary on Romans 1:7
 

Called to be saints - So called, or influenced by God who had called them, as to become saints. The word “saints,” ἅγιοι hagioi, means those who are holy, or those who are devoted or consecrated to God. The radical idea of the word is what is separated from a common to a sacred use, and answers to the Hebrew word, קדושׁ qadowsh. It is applied to any thing that is set apart to the service of God, to the temple, to the sacrifices, to the utensils about the temple, to the garments, etc. of the priests, and to the priests themselves. It was applied to the Jews as a people separated from other nations, and devoted or consecrated to God, while other nations were devoted to the service of idols. It is also applied to Christians, as being a people devoted or set apart to the service of God. The radical idea then, as applied to Christians, is, that “they are separated from other men, and other objects and pursuits, and consecrated to the service of God.” This is the special characteristic of the saints. And this characteristic the Roman Christians had shown. ~Barnes Notes Grace - This word properly means “favor.” It is very often used in the New Testament, and is employed in the sense of benignity or benevolence; felicity, or a prosperous state of affairs; the Christian religion, as the highest expression of the benevolence or favor of God; the happiness which Christianity confers on its friends in this and the future life; the apostolic office; charity, or alms; thanksgiving; joy, or pleasure; and the benefits produced on the Christian’s heart and life by religion - the grace of meekness, patience, charity, etc., “Schleusner.” In this place, and in similar places in the beginning of the apostolic epistles, it seems to be a word including all those blessings that are applicable to Christians in common; denoting an ardent wish that all the mercies and favors of God for time and eternity, blended under the general name grace, may be conferred on them. It is to be understood as connected with a word implying invocation. I pray, or I desire, that grace, etc. may be conferred on you. ~Barnes Notes

2] That bringeth salvation hath appeared to all men.

That bringeth salvation - Margin, to all men, hath appeared. That is, in the margin, “the grace which brings salvation to all men has been revealed.” The marginal reading is most in accordance with the Greek, though it will bear either construction. If that which is in the text be adopted, it means that the plan of salvation has been revealed to all classes of men; that is, that it is announced or revealed to all the race that they may be saved; compare the notes at Colossians 1:23. If the other rendering be adopted, it means that that plan was fitted to secure the salvation of all men; that none were excluded from the offer; that provision had been made for all, and all might come and be saved. Whichever interpretation be adopted, the sense here will not be essentially varied. It is, that the gospel was adapted to man as man, and therefore might include servants as well as masters; subjects, as well as kings; the poor, as well as the rich; the ignorant, as well as the learned. ~Barnes Notes

So what we have here is an introduction statement to the three verses to follow in this section of Chapter 2.  God has a plan.  He put us on this Earth to work out that plan.  He knows, as do we, that left alone, we would all perish in a second death.  He has brought us favor and pardon and put us into the greatest process ever conceived; the salvation process.  It is a part of His plan to offer this to everyone who has ever lived.

This is the fact and the lesson of verse 11.