For a long time, Hironaka's paper was probably the canonical reference. New proofs started to crop up in the late 90's, and truly changed the perception of the problem. The motivation was to make Hironaka's proof more constructive, and ultimately, make it tractable by computer algebra. This was ultimately done by Schicho and Bodnar in Maple (desing project). (Of course, the complexity is atrocious, but it still allows to run example beyond what one could reasonably do by hand.)
Two main groups worked on it in the late 90's, and they have been referenced already in the prior answers: Bierstone-Milman on the one hand, and Encinas-Villamayor and maybe a couple of others on the other hand (spanning many papers). There was a year-long seminar at Purdue in 2000-2001 led by Kenji Matsuki and Andrei Gabrielov to work through the Encinas-Villamayor proof. This was an opportunity to clarify quite a few things in the construction, and Matsuki posted his notes (128 p.) on Arxiv.
Ultimately, this lead Jaroslaw Wlodarczyk to offer the shortest proof of desingularization (by far, 24 pages vs. 100+ for everyone else) .
Simple Hironaka resolution in characteristic zero.
J. Amer. Math. Soc. 18 (2005), no. 4, 779--822
I hope this helps! (Sorry for the delay in my answer, I only joined MO last week-end.)
(Added later) How could I forget? There is also a nice expository paper that covers all these developments. It should be very useful for someone trying to understand how the various works fit together (does predate Wlodarczyk's paper though).
The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand).
Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 3, 323--403