"For as long as life is in me, And the breath of God is in my nostrils" (Job 27:3)
In Hebrew, teshuvah can be read as "return to the letter hey" (see Tanya, Igeret Hateshuvah, ch. 4). Since the letter hey is part of the Tetragrammaton (the four letters comprising the Divine Name), we can read teshuvah (conversion) as returning to HaShem. It's a great mystery but there you have it.
The genius of the Jerusalem Council was that they recognized that the non-Jews "returning" to G-d could not be excluded on the basis of physical conversion standards--that conversion was a spiritual dynamic occurring between a soul and its Source. C.K. Barrett observes something interesting about the language of conversion in the Apostolic Writings:
" [tois apo ton ethnon epistrephousin epi ton theon] [those among the Gentiles who are turning to God--Acts 15:19] could be said by a Jew of Gentile converts to Judaism; to a Jewish Christian the Christian conversion of Gentiles must have had to a considerable extent the same appearance. Gentiles were turning from whatever heathen gods they had previously worshipped to the God of the OT, the God of the Jews. It was this fact that gave strength to the requirement that they should behave like converts to Judaism, that is, should be circumcised and thereafter keep the Law. [epistrephousin--which is teshuvah or turning] could be used to describe what was required of Jews (3.19). [Acts 3:19 says "Repent, then, and turn to God, so that your sins may be wiped out, that times of refreshing may come from the Lord"]" pg. 728 of A Critical and Exegetical Commentary on the Acts of the Apostles by C.K. BarrettThis whole business of judging a convert based on his conversion papers is ridiculous. The thing that makes a convert is the pull of the Ruach. This is why the Council required only that the non-Jewish converts demonstrate their conversion by abstaining from "pollutions" of idols.
Teshuvah is simply turning from idolatry towards the One True G-d.