*This article is written from an Orthodox perspective and does not reflect the viewpoint of many modern Jews.*
Sexual relations between men are clearly forbidden by the Torah. (Lev. 18:22). Such acts are condemned in the strongest possible terms, as abhorrent. The only other sexual sin that is described in such strong terms is the sin of remarrying a woman you had divorced after she had been married to another man. (See Deut. 24:4). The sin is punishable by death (Lev. 20:13), as are the sins of adultery and incest.
It is important to note, however, that it is homosexual acts that are forbidden, not homosexual orientation. Judaism focuses on a person's actions rather than a person's desires. A man's desire to have sex with another man is not a sin, so long as he does not act upon that desire. In fact, Jewish tradition recognizes that a person who chooses not to do something because it is forbidden is worthy of more merit than someone who chooses not to do it because he doesn't feel like it; thus, a man who feels such desires but does not act upon them is worthy of more merit in that regard than a man who does not feel such desires.
I have seen one modern Orthodox source suggest that if homosexuality is truly something hardwired in the brain, as most gay activists suggest, then a man who acts upon that desire is not morally responsible for his actions, but I am not sure how wide-spread that opinion is.
Interestingly, female homosexual relations are not forbidden by the Torah. There is very little discussion of female homosexuality in the Talmud, and the few sources that mention it does not disqualify a woman from certain privileges of the priesthood, because it is "merely licentiousness," but there is a surprising lack of discussion of such issues as whether it would be grounds for divorcing a woman without her consent or without ketubah. Maimonides asserted that lesbian practices are forbidden because it was a "practice of Egypt" and because it constituted rebelliousness.
Sources: Judaism 101.