I converted to the Presbyterian faith from a Catholic upbringing a while back and have always wondered - but never had anyone to ask - why Catholics use "Trespassers" in the Lord's prayer (Please forgive our trespasses, as we forgive our Trespassers) and the Presbyterians use "Debtors". I'm sure someone has to have asked this question somewhere before me :-)
There's a great analysis by Dave Armstrong that goes into a lot more detail, but it comes from trying to best capture the point of the Lord's Prayer—as it's rendered in the Bible—in English.
There are two forms of the Lord's Prayer found in the Bible: one in Matthew and another in Luke.
In Matthew 6:9–13, the Lord's Prayer is rendered as (emphasis mine):
9“This is how you are to pray:
Our Father in heaven,
hallowed be your name,
10your kingdom come,
your will be done,
on earth as in heaven.
11Give us today our daily bread;
12and forgive us our debts,
as we forgive our debtors;
13and do not subject us to the final test,
but deliver us from the evil one.
But in Luke 11:2–4, it's rendered as (emphasis mine):
2He said to them, “When you pray, say:
Father, hallowed be your name,
your kingdom come.
3Give us each day our daily bread
4and forgive us our sins
for we ourselves forgive everyone in debt to us,
and do not subject us to the final test.”
Given it's more fleshed out, most Christians favor Matthew's version. In the original Greek, the word used in Matthew 6:12 is ὀφειλήματα, which means "debts". The word used in Luke 11:4 is ἁμαρτίας, which means "sins".
So, to be in line with the original manuscript, "debts" is the correct word, and indeed, most Christians (including all non-English Catholics) use "debts".
In English, however, "debts" and "debtors" usually refer to a loan: a monetary debt to a bank, for example. But later on in Matthew, it says:
14If you forgive others their transgressions1, your heavenly Father will forgive you
15But if you do not forgive others, neither will your Father forgive your transgressions1.
So while the actual word is "debts", it's clear the point of the Lord's Prayer is to say that one ought to forgive those who wrong us, just as God forgives us our sins and transgressions, not that we ought to forgive loans made upon us (i.e., God doesn't "loan" us sins).
To make clear the point of the prayer,"debts" was replaced with "trespasses". This translation became popular in 16th century England, even making it into the Book of Common Prayer. The Roman Catholic version of the Lord's Prayer is almost identical to the Book of Common Prayer's version at the time, and has not been changed since then.
Interestingly, the English version of the Missal recently underwent a fairly dramatic change as things were retranslated to better correspond to the original meaning of words, but the Lord's Prayer was skipped. I could not find any explicit justification for the omission, which suggests to me that the Vatican still considers the use of "trespasses" to accurately convey the intent of the prayer.
Note 1: Also translated as trespasses, depending on the translation.
Being raised Catholic, I was taught trespasses vs debtors. As a child, I was also taught to use the shorter version without:
For thine is the kingdom, and the power, and the glory, for ever and ever
When Freemasons say the Lord's prayer we use debtors. Which was the first time I had heard it done that way. I thought that we (Freemasons) were wrong until I researched and found out that it's very plausible that debtors is the correct translation.
My research showed that John Wycliffe in 1382, debts and debtors was used. Only the Tyndale version of 1525 used trespasses. I was completely perplexed! Ultimately we probably mean something like:
Forgive us our sins as we forgive those who sin against us.
But change is not something we do well :)
In line with what Mark Trapp said, please see this
So being christened United Methodist and confirmed United Presbyterian now (PCUSA) I have mostly seen debts. I have attended some Presbyterian churches that use sins instead of debts. I think its just all semantics. But like Peter Turner mentioned above, in the Latin, which is the official language of the Catholic Church and Vatican, you will notice it is debts:
Panem nostrum quotidianum da nobis hodie, et dimitte nobis debita nostra sicut et nos dimittimus debitoribus nostris.
The Greek is phrased slightly differently in Matthew, Mark, and Luke. (Another answer has already compared Matthew and Luke). Following the King James for the English translation:
και αφες ημιν τα οφειληματα ημων ως και ημεις αφιεμεν τοις οφειλεταις ημων
And forgive us our debts, as we forgive our debtors.
και οταν στηκητε προσευχομενοι αφιετε ει τι εχετε κατα τινος ινα και ο πατηρ υμων ο εν τοις ουρανοις αφη υμιν τα παραπτωματα υμων
And when ye stand praying, forgive, if ye have ought [i.e. anything] against any: that your Father also which is in heaven may forgive you your trespasses.
και αφες ημιν τας αμαρτιας ημων και γαρ αυτοι αφιεμεν παντι οφειλοντι ημιν και μη εισενεγκης ημας εις πειρασμον αλλα ρυσαι ημας απο του πονηρου
And forgive us our sins; for we also forgive every one that is indebted to us. And lead us not into temptation; but deliver us from evil.
An early Christian document dating from the first century - The Didache - prescribes the Lord's prayer to be said three times each day. The version that appears in The Didache uses the word for "debts" (οφειληματα) rather than trespasses (παραπτωματα). (It seems that there are two variants - one that contains the phrase "as we forgive our debtors", and another that does not). The Liturgy of St. James, which is the oldest eastern Liturgy still used also uses the term for "debt" rather than "trespass".
The Latin Vulgate versions of the Gospel texts seems to slightly blur the distinction that is in the Greek. In Matthew, debitum is used for ὀφείλημα ("debt"), with debenti being the "debtors"; in Mark, peccatum is used for παράπτωμα ("trespass"); in Luke, peccatum is also used for ἁμαρτία ("sin"), with debenti again being "every one that is indebted". The Douay-Rheims translation of these verses is:
And forgive us our debts, as we also forgive our debtors.
And when you shall stand to pray, forgive, if you have aught against any man: that your Father also, who is in heaven, may forgive you your sins.
And forgive us our sins, for we also forgive every one that is indebted to us. And lead us not into temptation.
In view of all this, and in support of what another has answered, I don't think there is a Catholic and non-Catholic wording issue here. There are Roman Catholic sources (e.g. Douay-Rheims Bible) that use "debts" and "sins" in place of "trespasses", and "debtors" and those "indebted to us" in place of "trespassers". It amounts to which underlying Greek source is implied and how the Greek and/or Latin is rendered into English by the translator.
The answer to this question, and it is a very good question, requires a deep understanding of the social, economic, and theological history of Europe. The key question to ask is the difference in meaning between debtors and trespasses.
Usury was against church law, and so Christians were barred from charging interest. Judaism had no such law, so European monarchs and rulers encouraged Jews to immigrate to their areas of control as far back at the 12th Century to better facilitate commerce, and so Jews became the banking class across Europe. Indeed, the Age of Exploration could not have happened but for the Jewish diaspora.
The Enlightenment and the rise of Liberalism brought about significant change, especially the dramatic rise of the Merchant Class. Usury laws were challenged to free the emerging industrial class in their pursuit of profit, and "debtors" was changed to "trespasses." The idea of forgiving debts was an anathema to Christians seeking monetary wealth, and the desire to protect and make sacred private property drove this change.
I am providing a very superficial account that led to the change, and it was a change to Protestantism in particular. One needs to understand the impact of the Black Death upon Europe, the impact of the Reformation, the impact of the Age of Exploration, the impact of the Enlightenment, and several other factors.
The bottom line is that the change was made to allow Protestants to increase their wealth. It's a fascinating evolution, but understanding requires a great deal of study, but study that is worth the effort.