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Avodah zarah is "idolatry." It literally means "strange worship" or "service." The second of the Ten Commandments forbids making and worshipping idols. From the moment that the Jews entered the the land of Israel, the pagan cults of the surrounding peoples were a periodic source of temptation, against which the prophets warned and fought.
The rabbis compiled a whole talmudic tractate about avodah zarah. Recognising the dangers of being a small, exiled, monotheistic people living among predominantly idolatrous neighbours, they expanded the range of restrictions on contact with idolatry.
It was forbidden to gain any economic benefit from idolatrous worship, from the verse "Nothing of it shall stick to your hand" (Deuteronomy 13:18). This is the basis of the recent sheitel controversy, which has led to some authorities banning wigs made from hair cut in religious ceremonies in India. (I am not competent to judge the halachic case for this, but one has to wonder about the investment of communal energy in this issue, when we are confronted with 50-per-cent intermarriage, world hunger, an imminently nuclear-armed Iran and other such challenges.)
Colloquially, avodah zarah is used to describe anything which is inappropriately worshipped, for example money, dieting, your mobile phone, or David Beckham.
I hear that the new craze in New York and London is yoga classes for dogs. At the end, the dogs are encouraged to clasp their paws in a gesture of prayer. I'm not sure if the dog can be said to be engaged in avodah zarah, but it seems suspiciously like worshipping your dog to me.