Most Americans probably don't realize that the newspaper ecosystem in the United Kingdom is quite a bit different than in the US. Except for the specialist-oriented Wall Street Journal and the McPaper, US newspapers are tied to a city or metropolitan region. In much smaller England, the national newspapers (which are all based in London) dominate. These are divided into three separate groups according to their perceived level of journalistic standards: quality, mid-market, and tabloid. Something else for Americans to note is an odd tradition (to this American) of having a mostly separate staff for the Sunday edition of the papers, most of which also have different names. In Scotland, Wales, and Northern Ireland, most of the English national papers run regional versions, and there are also a few regionally significant papers. Here's a rundown of the national titles ordered by circulation as of March 2011 (or Sunday circulation where appropriate).
Note 2011.07.15: A lot of people consider the now-defunct News of the World as the Sunday edition of The Sun. I've kept NotW separate because it is being wound up (or down), and the staff let go instead of being moved to The Sun.
Update 2011.07.15: Added the Morning Star, which is national but very small.