Aside from any awards being presented by other groups (such as the BSFA awards), and prizes given for things like the Masquerade, there are at least three awards that are commonly presented at modern Eastercons.
The Doc Weir AwardEdit
Voted on by the members of the current Eastercon, with the voting opening usually on the Friday of the convention and closing either late Saturday night or late Sunday night. Presented to a fan who has worked hard, over a period of years, to make things better for fandom (e.g. a con runner, an organiser of the Friends of Foundation, or just a hard working gopher), basically an unsung hero. The award is the possession of a silver cup which is engraved with the names of past winners, and which is passed on to the next winner the following Easter. Arthur Rose "Doc" Weir died March 4th 1961. The first Doc Weir award was presented in 1963.
More on the Doc Weir Award can be found on http://efanzines.com/DocWeir/index.htm including photos of Doc Weir, the cup and the award certificate.
The Ken McIntyre awardEdit
For fannish artwork (best "unpublished" artwork in the artshow, unpublished as in, not on a book cover, professional magazine cover etc. I believe convention publication and fanzine artwork is eligible as long as the original artwork it is on display in the convention artshow). I don't remember hearing about it for the last few years so I'm not sure what has happened to it. It was presented for the first time at the 1971 Eastercon and was still being presented at the 1999 Eastercon. I can find references leading up to the 2000 Eastercon requesting eligible submissions, but nothing after that. Ken McIntyre was a much-loved British fan artist who died of pancreatitis in August 1968.
The Phlosque awardEdit
An award presented by artist SMS to the piece of artwork (usually in the current Eastercon's artshow, but not necessarily!) which is the most sickeningly sweet, sentimental and schlocky. A unicorn painted on velvet would have a good chance of winning.
Details of Doc Weir and Ken McIntyre found on http://www.smithway.org/history/chap2b.html - Part of a history of fandom in the 60s compiled by Richard Lynch.