1 I exhort therefore, that, first of all, supplications, prayers, intercessions, and giving of thanks, be made for all men;
[AD 220] Tertullian on 1 Timothy 2:1
On the other hand, for our own petition, when we say, "Hallowed be Thy name," we pray this; that it may be hallowed in us who are in Him, as well in all others for whom the grace of God is still waiting; that we may obey this precept, too, in "praying for all," even for our personal enemies.

[AD 253] Origen of Alexandria on 1 Timothy 2:1
I think that supplication is a prayer offered with a special request for something a person lacks. From this is distinguished prayer which is more plainly offered with simple praise, not to obtain a request but simply to praise the nobility of great things. And I think that intercession is a petition for certain things addressed to God by someone who has some greater boldness, while thanksgiving is a statement of gratitude made with prayers for receiving good things from God, either when it is a great thing that is received and acknowledged with gratitude or when the greatness of the benefit is apparent only to the one who has benefited. ON PRAYER 14.2.Ambrosiaster: “Supplications” are on behalf of secular rulers, that they may have obedient subjects, and thus peace and tranquillity. “Prayers” are for those to whom power has been entrusted, that they may govern with justice and truth, so that all may prosper. “Intercessions” are for those in dire necessity, that they may find help. And “thanksgivings” refer to gratitude for God’s daily providences. Commentary on the First Letter to Timothy.
Theodore of Mopsuestia: Here supplications express the desire for good things from God. Prayers express the desire to be released from various evils. Intercessions ask for freedom from undeserved consequences. And thanksgivings express gratitude for blessings. Commentary on 1 Timothy.
Augustine: I prefer to understand by these words what the entire, or almost the entire, church observes: that we take as supplications those prayers which are said in celebrating the mysteries, before we begin to consecrate what lies on the table of the Lord. Prayers are said when it is blessed and sanctified and broken for distribution; and the whole church, for the most part, closes this complete petition with the Lord’s Prayer. The original Greek word helps us to understand this distinction: the Scripture seldom uses the word euche in the sense of oratio, but generally and much more frequently euchē means votum; whereas proseuchē,the word used in the passage we are treating, is always rendered by oratio.… Now, all the things which are offered to God are vowed, especially the oblation at the holy altar, for in this sacrament we show forth that supreme offering, by which we vow to abide in Christ, even to the union of the body of Christ. The outward sign of this is that “we, being many, are one bread, one body.” Consequently, I think that at this consecration and this preparation for Communion the apostle fittingly wishes that proseuchas, that is, prayers, should be made, or, as some have unskillfully rendered it, adoration, that is, what takes place at the offering, although this is more commonly expressed in Scripture by euchē. Intercessions, however, or, as your texts have it, requests, are offered while the blessing is being given to the people, for at that time, by the laying on of hands, the bishops, as intercessors, offer the members of their flock to the most merciful Power. When this is completed and all have received the holy sacrament, the whole is ended by thanksgiving, and this last is the very term called to our notice by the apostle. LETTERS 149.2.16.
John Cassian: “Supplication’ is a beseeching or petition for sins.… “Prayers” are those by which we offer a vow to God.… “Intercession” is customarily offered, in moments of fervor, for other men and women—our family, the peace of the world. To use St. Paul’s words, we pray “for all men, for kings and all in authority.” … “Thanksgiving” is when the mind recollects what God has done or is doing or looks forward to the good which he has prepared for those who love him, and so offers its gratitude in an indescribable transport of spirit. Sometimes it offers still deeper prayers of this sort; when the soul contemplates in singleness of heart the reward of the saints and so is moved in its happiness to pour forth a wordless thanksgiving.

[AD 384] Ambrosiaster on 1 Timothy 2:1
“Supplications” are on behalf of secular rulers, that they may have obedient subjects, and thus peace and tranquillity. “Prayers” are for those to whom power has been entrusted, that they may govern with justice and truth, so that all may prosper. “Intercessions” are for those in dire necessity, that they may find help. And “thanksgivings” refer to gratitude for God’s daily providences. .
[AD 407] John Chrysostom on 1 Timothy 2:1
Let us then exhort the saints to give thanks for us. And let us exhort one another toward gratitude. To ministers especially this good work belongs, since it is an exceeding privilege. Drawing near to God, we give thanks for the whole world and the good things we commonly share. The blessings of God are shared in common, and in this common preservation you yourselves are included. Consequently, you both owe common thanksgivings for your own peculiar blessings and for those shared in common with others, for which you rightly should offer your own special form of praise.… So then let us give thanks also for the faith that others have toward God. This custom is an ancient one, planted in the church from the beginning. Thus Paul also gives thanks for the Romans, for the Corinthians and for the whole world.

[AD 407] John Chrysostom on 1 Timothy 2:1
The Priest is the common father, as it were, of all the world; it is proper therefore that he should care for all, even as God, Whom he serves. For this reason he says, "I exhort therefore that, first of all, supplications, prayers, intercessions, and giving of thanks be made for all men." From this, two advantages result. First, hatred towards those who are without is done away; for no one can feel hatred towards those for whom he prays: and they again are made better by the prayers that are offered for them, and by losing their ferocious disposition towards us. For nothing is so apt to draw men under teaching, as to love, and be loved. Think what it was for those who persecuted, scourged, banished, and slaughtered the Christians, to hear that those whom they treated so barbarously offered fervent prayers to God for them. Observe how he wishes a Christian to be superior to all ill-treatment. As a father who was struck on the face by a little child which he was carrying, would not lose anything of his affection for it; so we ought not to abate in our good will towards those who are without, even when we are stricken by them. What is "first of all"? It means in the daily Service; and the initiated know how this is done every day both in the evening and the morning, how we offer prayers for the whole world, for kings and all that are in authority.

[AD 430] Augustine of Hippo on 1 Timothy 2:1
I prefer to understand by these words what the entire, or almost the entire, church observes: that we take as supplications those prayers which are said in celebrating the mysteries, before we begin to consecrate what lies on the table of the Lord. Prayers are said when it is blessed and sanctified and broken for distribution; and the whole church, for the most part, closes this complete petition with the Lord’s Prayer. The original Greek word helps us to understand this distinction: the Scripture seldom uses the word euche in the sense of oratio, but generally and much more frequently euch&#; means votum; whereas proseuch&#;,the word used in the passage we are treating, is always rendered by oratio. … Now, all the things which are offered to God are vowed, especially the oblation at the holy altar, for in this sacrament we show forth that supreme offering, by which we vow to abide in Christ, even to the union of the body of Christ. The outward sign of this is that “we, being many, are one bread, one body.” Consequently, I think that at this consecration and this preparation for Communion the apostle fittingly wishes that proseuchas, that is, prayers, should be made, or, as some have unskillfully rendered it, adoration, that is, what takes place at the offering, although this is more commonly expressed in Scripture by euch&#;.Intercessions, however, or, as your texts have it, requests, are offered while the blessing is being given to the people, for at that time, by the laying on of hands, the bishops, as intercessors, offer the members of their flock to the most merciful Power. When this is completed and all have received the holy sacrament, the whole is ended by thanksgiving, and this last is the very term called to our notice by the apostle. .
[AD 435] John Cassian on 1 Timothy 2:1
“Supplication’ is a beseeching or petition for sins…. “Prayers” are those by which we offer a vow to God…. “Intercession” is customarily offered, in moments of fervor, for other men and women—our family, the peace of the world. To use St. Paul’s words, we pray “for all men, for kings and all in authority.” … “Thanksgiving” is when the mind recollects what God has done or is doing or looks forward to the good which he has prepared for those who love him, and so offers its gratitude in an indescribable transport of spirit. Sometimes it offers still deeper prayers of this sort; when the soul contemplates in singleness of heart the reward of the saints and so is moved in its happiness to pour forth a wordless thanksgiving.
[AD 990] Oecumenius on 1 Timothy 2:1-2
First of all, then, I urge that petitions, prayers, intercessions, and thanksgivings be made for all people; for kings and all who are in high positions, that we may lead a peaceful and quiet life in all godliness and dignity.

In daily worship, this would be before the remembrance of others, he says. To make petitions. A petition is a plea offered for deliverance from troubles. Prayer is a request for good things. Intercession is an accusation against wrongdoers.

thanksgivings, for it is necessary to give thanks for the good things that happen to one another.

be made for all people. Since the bishop is a common father, he ought to pray for all people, both the faithful and the unbelievers, friends and enemies, those who slander and those who afflict.

for kings. Then, so that the matter might not appear to be flattery, first saying, be made for all people; thus he declared, "for kings and all who are in high positions."

that we may lead a peaceful and quiet life. Let no Christian be disturbed in any way; therefore, during the time of the mysteries, let him remember that the king, since he is often unfaithful, shows that our profit is their salvation. But how is their salvation our profit? How? For if they are preserved and conduct themselves excellently in wars against enemies, we live quietly and peacefully with no one disturbing us. It would indeed be absurd if they were to be exhausted for the common safety, but we do not even bring together that which is in us as help (I mean prayer) for them.

in all godliness. There are three kinds of wars: one is against the Barbarians; one is against those of the same race; the third, which the soul wages against the body, which is the hardest of all. For the war against the Barbarians brings only death or slavery. The war against those of the same race, I have learned to avoid through gentleness. "Instead of loving me," he says, "they attacked me, but I prayed." (Ps. 109:4) And, "I was peaceful among those who hate peace". (Ps. 120:6) But the one within ourselves is difficult to accomplish, and it brings harm to the soul. Therefore, he wishes to pray for the kings, so that they may both conquer and endure, knowing that the peace from the Barbarians greatly contributes to the peace within ourselves. For if one must be at leisure and know God, then freedom and harmony are necessary for the knowledge of God and kindness. How then could those engaged in wars with the Barbarians be at rest? For it is noble and undeniable to pray for all people, both Greeks and others, so that they may turn back.