Western Buddhism is a freethinking, self-empowering strain of Buddhism which was born in the 19th century Transcendentalism and Theosophy movements, and given new life in the 20th century by D.T. Suzuki's Zen lectures in the 1950s. It is about as mainstream as Buddhism can be, with its own fine, rationalist publisher (Shambala).
Western Buddhism respects all of the philosophy of Buddhism, and generally reveres Buddhist leaders such as the Dalai Lama, but Western Buddhists are unattached to the traditions of Theravada, Mahayana and Vajrayana. The large numbers of agnostic, atheist, and otherwise self-reliant Western Buddhists have made its disattachment from tradition quite secure. That means that calling yourself a "Western Buddhist" is a declaration of faith in the Buddha, dharma and sangha, and not a statement of piety for any particular church (as tempting as it might be to devote oneself to a particular branch of Buddhism). You will be evaluated by your words and actions and not by how often you come to temple.
A brief history of American experience with Buddhism
Buddhism's initial reception in the United States was orientalist, an unscientific and unspiritual approach which doesn't mean much historically. It seems pretty sensible, in retrospect, that the initial reaction to Buddhism would be to compare it to elements of Christianity, or denegrate it as idolatry or nihilism. In fact, these are the initial reactions of some Christians even today.
Western Buddhism gained its own life only when people began to consider it for its own merits, rather than lining it up alongside Christianity and apologizing for its faults.
I didn't write this yet but I will get around to it someday.