Praying to people outside the Trinity

A general rule of thumb would be that those Christian bodies that claim apostolic succession (think Orthodox, Coptic, Jacobite, Ethiopian Orthodox, Roman Catholic) and can demonstrate that succession with a distinct lineage BY NAME from apostle to bishop to bishop (and so on) generally ask for the intercessions of those who are alive in Christ (He is the God of the living, not of the dead). Note that these Christian bodies are also the oldest in the world.

Those denominations that have a much shorter history (usually dating from the 16th century at the earliest) and can in general demonstrate no apostolic succession (and generally do not believe it to be important) tend to pray to God alone. Note that these denominations are the newest in the world.

Some Christians do not agree with the teaching of the Catholic Church on when the judgement occurs. Where Catholics believe that some who have died have already received final judgement, and been accepted into heaven, and thus are risen from the dead, and alive, some Christians believe that judgement has not occurred, and that departed persons have not yet been judged, and are not yet in heaven, and thus cannot hear or act on requests for intercession.

On the other hand, some Lutherans, and most Anglicans accept that at least some Christians who have died have already been judged worthy, are risen from the dead, and that the intercessions of these persons, may be asked by those living on earth. So one might ask the Mother of God, or St. Luke, or even Great Grandma Smith, to pray for us, and they will do so, just as one might ask the next door neighbor to pray for us in time of sickness.

As stated above by others no Catholic or Orthodox prays to a saint in the same way that he or she prays to the Blessed Trinity. In Latin for instance two entirely different words are used for the two actions. On Earth I would ask my brothers and sisters to pray for me in my hour of need. We simply believe that those in Heaven can still pray for us so we ask for their prayers. We understand that they have no power, they are just fellow Christian who have entered into glory. If we did pray to saints as we pray to God then we would not be Christians but polytheists. It is Christ and Christ alone in whom we place our trust.

Catholics pray for the entire communion of saints to intercede on their behalf.

It's more than just the Latin Rite of the Catholic Church (what most people refer to as Roman Catholic) it's the entire Church in communion with the pope, as well as the Eastern Orthodox Church who prays fervently to the saint.

According to Catholic doctrine Angels are also saints (St. Raphael, St. Michael) and anyone who believes in angels must believe they can intercede for them (what else would they do).

I don't know whether it is common for other Christian denominations to pray to their guardian angels, but the belief in their intercessory powers is certainly there, at least in popular culture and literature.

First, it's necessary to understand that Catholics (whether Roman or Eastern Catholic, or Eastern Orthodox) do not pray to the saints; rather, they ask the faithful departed to pray for them to God.⁽¹⁾

The idea has its basis in the "communion of saints," which may be derived from Heb. 12:1 (as well as other scriptures):

Therefore, since we are surrounded by so great a cloud of witnesses, let us rid ourselves of every burden and sin that clings to us and persevere in running the race that lies before us (NABRE)

Continuing from Heb. 11, the "cloud of witnesses" referred to is the faithful who have already died in Christ (those of the Old Covenant looking forward to Christ and the cross, those of the New Covenant looking back).

Since those who have died in Christ are now alive in him, the Catholic understanding is that we can converse with them just as with any believer still alive in this realm. Since they are alive we can asked them to pray for us just as we would approach a fellow believer who is living in this realm to pray for us: "Bob, I am really struggling with X right now. Would you please pray for me to find victory in this area?"

Coupled with the idea that "The fervent prayer of a righteous person is very powerful" (Jam. 5:16, NABRE), it then seems to make sense to ask those who have been recognized by church authority as being righteous and exemplary to pray for you.

The logic is, then, that if (a) you accept that asking another believer to pray for you is okay, and (b) you believe that those who have died with Christ are alive with him, then it follows that you can ask those who have died in Christ to pray for you.

Footnotes

⁽¹⁾ It's not uncommon for individuals to actually, and wrongly, pray to the saint; but if you pay attention to the liturgy the words are of the form "St. Peter, pray for us; St. Paul, pray for us..."

The phrase "pray to" confuses most people as it gives the impression that anyone who asks a saint to "pray for" them is actually praying to the saint as if he/she was God. This isn't true and is probably an artifact of language.

The word "pray" also has the meaning to "make an earnest petition" so Catholics, and, I think, other Christians, call upon any saint, past or present, to pray for us.

Just as I might ask my friends to pray for me I might also ask a saint in Heaven to pray for me.