I think there's been some sort of market since 1996, but the liquid ones really started in the 2002 mid-terms.
Check out
Intrade. Intrade is a spread betting market, so if you're familiar with these feel free to ignore the next 2 paras.
Any individual contract gives a probability of a certain event happening. Of course this means that the opposite could happen, but Intrade got most (more than half, IIRC) of the 2006 mid-terms right (i.e. if a candidate had more than 50% chance of winning on InTrade, he usually won). Obama's chance of winning the election is 61%, which is pretty high.
On InTrade, you basically buy a contract at whatever price and earn or lose the difference. Say I buy 1 contract on Obama to win the election at 61% ($6.10). The probability goes up to 63%. I've made 2%, which on InTrade would be $0.20.
There's also the Iowa Electronic Markets, which also has binary and vote share contracts (Obama to win, 58%). The paper cited in this
article has more if you can find it. I think you can also spread bet on the elections at UK sites like IG Index, Cantor Index and City Index.