Let $X$ be a smooth variety and consider the diagonal $\Delta \subseteq X \times X$. It seems to be well-known that the exceptional divisor in the blow-up of $X \times X$ along $\Delta$ is isomorphic to the projectivized tangent bundle $\mathbb P(\mathcal T_X)$ of $X$ but I can't find a reference or a proof; where might I find one?
Chuck, I think you can find references for both of the statements 1,2 that Allen gives in Fulton's Intersection theory, Appendix B, Section 5, and maybe also section 7. There, I think (1) is proved more generally for a regularly embedded subscheme. I asked a question about (2) some time ago. You might find what you are looking for (including how one proves 2, which is easy, and "intuition") here:
Fulton doesn't include proofs for everything but gives references to EGA. (Since I don't have the book with me, the section numbers might be a little off, maybe someone can correct me.)