All the previous answers are quite good but may be confusing taken all together; I'll try to organize and expand on the info just a bit.
1. Unitarianism started as a movement within Christianity, part of the Reformation; it was as biblically based as any Protestant church or movement, more so than some, denying the doctrine of the divinity of Jesus (and thus the Trinity) precisely because it couldn't find it literally stated in the Bible. This resulted in the Unitarians being persecuted by Protestants and Catholics alike, until people learned to give up religious persecution altogether.
2. In North America, many of the old Puritan churches (and some others) became Unitarian around the beginning of the 19th century, again first by simply studying the Bible and finding that the Trinity wasn't in it. Gradually they concluded, from the fact that neither side could prove its teachings conclusively from the Bible, that the Bible wasn't meant to be used this way at all; they became increasingly free-thinking in other ways too, and socially progressive, finally evolving beyond Christianity altogether. (Individual members may identify as Christian, but the movement as a whole does not.) In 1961 these churches merged with the Universalists - a different movement, believing in universal salvation, which had been moving in a similar direction - to form the UUA, which you discovered on the web. (Canadian Unitarians and Universalists were part of it at first but within the past decade broke away to form their own national organization.)
3. In Europe, the Unitarian churches generally evolved towards liberalism and non-credalism, like here, but remained Christian; the UUA has very good relations with them. I've met Unitarian ministers from England and Transylvania; they use more Christian language than we UUs in the States would be comfortable with, but in all other respects we are much the same, free-thinking and socially liberal. The Irish church you found is in this category.
4. BUT there are still people around who identify as Unitarians in the original Christian sense, who are Biblical literalists and social conservatives. Not many, as far as I know, but some. They strongly object to being confused with us Unitarians of the UUA, whom they consider heretics (a term many of us in fact proudly accept).
牛顿童鞋隐藏的就是深。
