How it has been translated into English
The Greek word is a form of ὀφείλημα (3783), which according to Strong's has been translated
1) that which is owed 1a) that which is justly or legally due, a debt 2) metaph. offence, sin
The word comes from ὀφείλω (3784):
1) to owe 1a) to owe money, be in debt for 1a1) that which is due, the debt 2) metaph. the goodwill due
According to BDAG (and Moulton & Milligan), the primary meaning of ὀφείλημα is
that which is owed in a financial sense, debt, one’s due.1
It can also refer to an "obligation in a moral sense, debt" (and is used in a similar way to the Aramaic חוֹבָא in rabbinical literature).2
Analysis of usage in biblical literature
The other appearances of ὀφείλημα in biblical literature support the primary meaning of this noun according to BDAG.
Ἐὰν ὀφείλημα ἦ ἐν τῷ πλησίον σου, ὀφείλημα ὁτιοῦν, οὐκ εἰσελεύσῃ εἰς τὴν οἰκίαν αὐτοῦ ἐνεχυράσαι τὸ ἐνέχυρον (LXX, emphasis mine).3
If there is a debt with your neighbor, whatever kind of debt, and you shall enter into his house to take in pledge his pledge.4
καὶ πᾶσαν διάνοιαν μεταστρέφει εἰς εὐωχίαν καὶ εὐφροσύνην καὶ οὐ μέμνηται πᾶσαν λύπην καὶ πᾶν ὀφείλημα (LXX, emphasis mine).5
It turns every thought to feasting and mirth, and forgets all sorrow and debt.6
καὶ πᾶν ὀφείλημα βασιλικὸν καὶ τὰ ἐσόμενα βασιλικὰ ἀπὸ τοῦ νῦν καὶ εἰς τὸν ἅπαντα χρόνον ἀφιέσθω σοι (LXX, emphasis mine).7
Every debt you owe to the royal treasury and any such future [debts] shall be canceled for you from henceforth and for all time.8
τῷ δὲ ἐργαζομένῳ ὁ μισθὸς οὐ λογίζεται κατὰ χάριν ἀλλὰ κατὰ ὀφείλημα... (NA27, emphasis mine).9
Now to one who works, wages are not reckoned as a gift but as something due.10
This last use by Paul of Tarsus in his letter to the Romans makes it clear that in the first century (the same approximate historical time frame in which the Gospel commonly attributed to Matthew was likely written), the meaning of ὀφείλημα was in contradistinction to a gift (χάριν).
Analysis of usage in extrabiblical literature
The distinction between χάριν and ὀφείλημα brought out by Paul of Tarsus is elucidated several hundred years earlier by Thucydides (4th-5th century BCE), when he writes
οὐκ ἐς χάριν, ἀλλʼ ἐς ὀφείλημα
not as a favor but as payment of an obligation.11
The primary meaning as given by BDAG is supported by numerous other extrabiblical writings as well.12
Both Matthew Black and Bauer, Danker, & Arndt suggest that ὀφειλήματα means 'sins' in Matthew 6:12.2 The parallel reading in Luke 11:4 has "τὰς ἁμαρτίας ἡμῶν" ('our sins') which lends support to this reading. In addition, analysis of the verb form of this noun (ὀφείλω)13 and its relationship to חוֹבָא and corresponding חַיָּב in rabbinical literature also lend support to this reading (particularly if the prayer was originally composed in Aramaic).
The strongest support, however, comes from the immediate context of the prayer, recorded in vv. 14-15:
Ἐὰν γὰρ ἀφῆτε τοῖς ἀνθρώποις τὰ παραπτώματα αὐτῶν, ἀφήσει καὶ ὑμῖν ὁ πατὴρ ὑμῶν ὁ οὐράνιος· ἐὰν δὲ μὴ ἀφῆτε τοῖς ἀνθρώποις, οὐδὲ ὁ πατὴρ ὑμῶν ἀφήσει τὰ παραπτώματα ὑμῶν.14
This context makes a clear comparison between ὀφειλήματα and παραπτώματα, the latter meaning "a violation of moral standards, offense, wrongdoing, sin" (clearly not solely restricted to financial debt).15
For these reasons (but particularly the contextual support), it is likely that 'sins' are the intended 'debts' in this context, but the actual reading is that word generally used for financial 'debts', i.e. 'that which is owed.'
1 William Arndt, Frederick W. Danker, and Walter Bauer, A Greek-English Lexicon of the New Testament and Other Early Christian Literature (Chicago: University of Chicago Press, 2000), 743.
2 Matthew Black, An Aramaic Approach to the Gospels and Acts, 3rd ed., ed. Patrick H. Alexander (Peabody, Massachusetts: Hendrickson Publishers, 1998), 67, 140.
3 Septuaginta: With Morphology, electronic ed. (Stuttgart: Deutsche Bibelgesellschaft, 1979), Dt 24:10.
4 Rick Brannan, Ken M. Penner, Israel Loken, et al., eds., The Lexham English Septuagint (Bellingham, WA: Lexham Press, 2012), Dt 24:10.
5 Septuaginta, Esd A 3:20.
6 The Holy Bible: New Revised Standard Version (NRSV) (Nashville: Thomas Nelson Publishers, 1989), 1 Esd 3:20.
7 Septuaginta, 1 Mac 15:8.
8 NRSV, 1 Mac 15:8.
9 Eberhard Nestle, Erwin Nestle, Barbara Aland, et al., The Greek New Testament, 27th ed. (Stuttgart: Deutsche Bibelgesellschaft, 1993), 415.
10 NRSV, Ro 4:4.
11 cp. Thu. 2, 40, 4. From BDAG, 743.
12 "Pla., Leg. 4, 717b; Aristot., EN 8, 15, 1162b, 28; 9, 2, 1165a, 3; SIG 1108, 10 [III/II B.C.]; PHib 42, 10 [262 B.C.]; PLond III, 1203, 4 p. 10 [113 B.C.]; POxy 494, 10 ὀφειλήματα." Ibid., 743. Also cf. Moulton and Milligan, 468.
13 Cf. especially the verbal form (ὀφείλετε) in Romans 13:8.
14 Nestle & Aland, 27th ed., 13-4.
15 BDAG, 770.