Best mathematics conference?  One conference per answer. Explain why it was great (mathematically or otherwise), and preferably post a link to the conference website or to abstracts/proceedings. 
 A: The Talbot workshops are mostly aimed at graduate students and early professionals.  The topics and plenary speakers vary year-to-year, but the ones I have attended were all very illuminating.  The 2009 workshop, on the Fukaya category, was especially good, and there are notes from each talk on the web page linked above.
A: I'll start. The New Zealand Institute of Mathematics holds a conference every year, a different location and topic each time. In 2006, at Taipa, the topic was geometric topology. Jeff Cheeger gave an excellent plenary series of talks on spaces of Einstein metrics. He and Tian proved certain collapsing results for these metrics, laying the framework for compactifying moduli spaces of Einstein metrics.  There were two talks in the morning, and one in the evening after dinner, leaving the afternoon free for activities such as windsurfing. If you can make it, I recommend it. This year (January 2010) will be on quantum topology (although I have to say I won't be able to go). 
A: The famous AMS Summer Research Institute on Automorphic Representations and L-functions held at Corvallis in 1977. It was organized by Borel, Casselman, Deligne, Jacquet, Langlands, and Tate to give a coherent account of the state of the Langlands program. Beyond the organizers, speakers included Springer, Tits, Cartier, Lusztig, Piatetski-Shapiro and Arthur. Generally, speakers were assigned topics to talk on.  The proceedings have been a wonderful resource during the thirty years since they were published. They are available on the AMS website (http://www.ams.org/online_bks/online_author.html#B).
A: Now that this is turned into a community wiki post, I'll throw an answer out there: No, a non-answer!
And my non-answer is this: There are too many conferences, and the vast majority of them too specialized for most of us, to answer this question in a way that would be useful to most mathematicians. (Yeah, I am sticking my neck out here, speaking for “most mathematicians” when it's not altogether clear that I am competent to speak for myself.)
I tend to prefer narrowly focused conferences myself, in a field that I actually know something about. That way, I can at least feel appropriately ashamed when I can't follow a talk on a subject I had thought I understood. Which provides an incentive to do something about it. So if I may add to the original question, then? Why do people go to big general conferences? What do they get out of it? Okay, this may sound too negative. Breadth is good, and I wish I had more of it, but mathematics is a vast field and only a few of us can attain deep understanding in more than a couple of subfields, at best.
A: The 1900's Paris ICM... just because of Hilbert's talk, in which he modelled the shape of a rather big amount of the mathematics that were to come; and not just by becoming fancy, but by identifying really difficult and interesting problems and, even more importantly, by pointing to logic and rigor so that others could see the necessity for a thorough cleaning of their mathematical building.
(Sorry, I could not locate any link to the proceedings ;P)
A: The Solvay Conference (see http://en.wikipedia.org/wiki/Solvay_Conference), although formally being a physics conference, was attended by Poincare and was fundamental to establishing quantum mechanics and general relativity as important topics in mathematical physics.
