Known bounds and values for Ramsey Numbers

The best resource online is the (frequently updated) survey Small Ramsey Numbers by Stanisław Radziszowski, in The Electronic Journal of Combinatorics. Go to http://www.combinatorics.org/ and click on the link for "dynamic surveys". When I first wrote this answer (December 12, 2010), the survey version dated August 2009. As of this edit (June 8, 2014), the most recent update on the Ramsey numbers paper is dated January 12, 2014. The paper gives extensive references where one can find complete proofs or details of the computations involved.

Radziszowski himself is responsible for several improvements to the bounds listed there, and you may want to check his page for recent results not yet included. Although the emphasis of the paper is on exact values, it also includes references for asymptotics and general upper bounds. With respect to the latter, there have been significant recent advances (particularly, by Conlon and his collaborators), and you may want to check the pages of the authors listed on page 9 of the survey, for possible improvements.

I found through another answer in this site a link to Geoffrey Exoo's page (somewhat under construction, it seems), which contains additional improvements due to Exoo (mostly unpublished).