From First and Second Timothy, Titus by George T. Montague, SM, commenting on 1 Timothy 2:1-8:
There are only slight differences in the four types of prayers listed. Supplications are prayers occasioned by some concrete circumstances or pressing need. Prayers, frequently associated with supplications, is a more general term for prayer. It was in answer to his disciples’ request to “teach us to pray” that Jesus taught them the Lord’s Prayer (Luke 11:1–13). Petitions, sometimes translated “intercessions,” comes from a verb used to describe the favor a subject enjoys in being admitted to the presence of the king so as to present a request. For the Christian this means to beseech the “king of ages,” whom Paul just mentioned (1 Tim 1:17). The series climaxes (with no connecting “and” in the Greek) with thanksgivings.
A plural word for “Eucharist,” here the meaning is simply expressions of gratitude for past benefits bestowed on those for whom prayers are offered. In Phil 4:6 Paul tied three of these terms (prayers, supplications, and thanksgivings) together, suggesting in both places that thanksgiving should always accompany what Christians ask for in prayer.
This prayer is not just for one’s family or the local congregation or even for the Church. It is for everyone. Christians considered prayer for one another and for the Church a sacred duty (Acts 12:5; James 5:14–15). This was not difficult and sprang spontaneously from the Jewish soul schooled in the value of intercessory prayer. But Jesus had also told his disciples to pray for their enemies (Matt 5:44), and the Church could never forget that he himself had done so as he was dying on the cross (Luke 23:34), thus fulfilling the prophecy of the suffering Servant (Isa 53:12). The Christian’s prayer must extend as far as his charity (Matt 5:44). Luke, who records Jesus’ prayer for his enemies from the cross, also tells us that Stephen prayed for those who stoned him, and therefore for Paul himself, who approved of his execution (Acts 7:58–60).