The main issues surrounding whether homosexuality is permitted or condemned in various Christian denominations are based on passages in Leviticus and Romans, and more generally on theories of Biblical interpretation. At the present, Roman Catholics and many evangelical denominations consider homosexuality a sin, but many mainstream Protestant denominations including the Episcopal Church, some Lutheran synods, the United Church of Canada and some Presbyterians have gay priests and perform gay marriages.
Although Leviticus forbids certain homosexual acts, this prohibition occurs in the context of a series of ritual purity laws. Since Christians believe that the New Covenant established by Christ abrogated the ritual laws of the Old Testament (e.g. Christians can eat shellfish, mix meat and dairy, etc.) many Christians do not see Leviticus as binding.
St. Paul, in the Epistle to the Romans, condemns some homosexual acts, but we are not quite sure which ones or why, and thus it is somewhat hard to apply this as a standard.
Given how little the Bible has to say about homosexuality, many Christians think that issues like whether a relationship is monogamous, loving, and a good environment for raising children is more important than whether it is homosexual or heterosexual.
Romans 1:26-27 ~ Because of this, God gave them over to shameful lusts. Even their women exchanged natural relations for unnatural ones. In the same way the men also abandoned natural relations with women and were inflamed with lust for one another. Men committed indecent acts with other men, and received in themselves the due penalty for their perversion.