This is way too long of an essay to include as a reference on the appropriate page, so here goes. For those who somehow don't know, we're discussing here how we know Dolores Umbridge is pure-blood.
Well, we actually don't know anything - J. K. Rowling often surprises us with this sort of thing. But we've got a lot more evidence for this than for other things stated on here.
First and foremost: the truly, irrefutably canon bit: the WOMBAT Test from J. K. Rowling's Official Site, archived at the Harry Potter Wiki at the link given, states that all Wizengamot members must provide evidence of pure-blood status. Umbridge was on the Wizengamot as of 1995, and therefore had to have shown pure-blood credentials.
We know she was either half-blood or pure-blood, or else she would have been imprisoned by the Muggle-Born Registration Commission. Cornelius Fudge, her employer, despised anyone who was pure-blood or less, as revealed by a comment made by Albus Dumbledore in June of 1995, and not only did he respect and employ her, but he honoured her and allowed her to lead him!
Also, make note how at a time when pure-bloods were treasured and anything less either disrespected or outright despised, Umbridge was placed in charge of the centre of the anti-Muggle-born propaganda, the Muggle-Born Registration Commission, entrusted with the most important job, placed at the top of the hierarchy. If she had even one Muggle or Muggle-born relative, it is almost certain more pure-blooded, elder members of the Ministry or Death Eater organisation would have been put in charge instead.
We also know that she is related to the Selwyns, a family that was so pure-blood they were included in the "Sacred Twenty-Eight" list. Yaxley, who was also there and related to the Selwyns, Blacks and several other pure-blood families, would certainly not have hesitated to call her on it if she was lying about her relation or if they weren't actually pure-bloods.
All in all, I think it is rather reasonable to state Umbridge was a pure-blood.
Note from 6 December, 2013: Shocking pink conveys a certain punky 'yes I've got a Muggle-born father and I'm not ashamed of it' attitude, and Umbridge certainly wears a lot of that colour. Perhaps this is a nod towards her parentage?