Shirk, for the purposes of this article, is the Islamic concept of the sin of idolatry. It is the vice opposed to the virtue of tawhid (also, tawheed), the Islamic word for monotheism.
It derives from the Arabic word for partnership, and that root meaning informs the Islamic concept. One commits shirk by associating some lesser being with Allah. This sin is committed if you imagine that there is some lesser spirit than Allah who is also worthy of worship or veneration, or to whom you can pray to intercede with you rather than taking your case to Allah Himself. This sin is also committed if you imagine that Allah is associated with some artifact here on earth, or that some such object is worthy of worship or veneration.
Islamic monotheism requires the Muslim instead to revere God as the unique creator and sustainer of the universe (tawhid ar-rububiyya, the unity of creation). This condemns as shirk the concepts of polytheism and atheism, the idea that there may be many gods, or no god.
Islamic monotheism also requires the faithful to believe that God is the only being entitled to worship, praise, or to be associated with His names and attributes (tawhid al-asma wa as-sifat, unity of the names and attributes). This condemns as shirk, first, the notion that some human being or other being beneath Allah is worthy of reverence; or that God possesses anthropomorphic qualities.
Islamic monotheism also requires the faithful to practice unity of worship (tawhid al-ibada). This condemns as shirk any prayer addressed to some other deity (outward shirk). It also requires God to be the true object of worship, and as such condemns as shirk acts of worship or piety whose actual goal is pride, caprice, or seeking admiration rather than focus on Allah (inward shirk).
Islamic theologians have thought hard about the implications of monotheism. The lessons they have deduced, and their insistence on the absolute unity and transcendence of Allah are worthy of consideration by all of the Abrahamic religions.