Picking up on some of the discussions recently, which religions are Avodah Zarah and why? How does that translate into our relationships with them?
Well it's a question of which religions attribute G-d-like power to physical objects.
Most often it comes up in questions regarding prohibited benefit; for instance a few years ago there was concern that hair for wigs was coming from Hindu rituals. If Hinduism is "avoda zara", then that hair may be prohibited from benefit. (In this case I believe the conclusion reached was that Hinduism as practiced in India is considered avoda zara, but for complex reasons, the hair is not prohibited.)
I've heard different things about Buddhism, in our categorical worldview it may be avoda zara or it may in fact be atheism. (For the little I understand of Eastern practices, I'd assume Shinto rituals are less problematic still.)
Islam, the Bahai faith, Jainism -- all monotheistic, nothing to talk about.
Basically today it only comes up with questions of prohibitions such as entering a shrine, or if a ritual item is prohibited from benefit. Philosophically beyond that ... well our primary focus is on improving our own roles as Jews. It appears that good fences make good neighbors, to a degree.
One would have to look at Wikipedia or elsewhere to see if they worship One God (good) or intermediaries (probably avodah zara) or many gods (definitely idolatry). Some Eastern religions don't have a strong stance on the issue of God, so they may not be avodah zara, though many followers are likely atheists.
Islam is not avodah zarah. Hinduism is. Folk religions in Africa and elsewhere are mostly avodah zara. Buddhism is a philosophy with many varieties so it would depend. Shintosim seems to be avodah zara while Sikhism seems not to be.
See Hilchos Avoda Zara (1:2, 2:1-2).