A new buzzword in American politics. Clean elections refers to a new policy in campaign finance wherein candidates are financed by taxpayer money, instead of by private corporations and special interest groups. The upshot of this is that it gives candidates outside the two major parties who are still able to drum up enough support for their candidacy the opportunity to receive funding to carry on a decent campaign.
If the system were ideally implemented, candidates would be required to use only public funds to conduct their campaign, and would be required to turn down donations from outside interests. This would limit the ability of the two major parties to stifle the election by having an inordinately larger amount of money. It's also meant to restore some public trust in a governmental system which has been less than trustworthy at times, to say the least. By certifying that a candidate is "clean", the candidate is also saying that he is not indebted to private interests, so the voters can trust that the candidate's ideas and policies are his (or hers) alone, not influenced by private interests.
Clean elections are being phased in slowly in a few states. My home state, Maine, recently passed a clean elections referendum. Unfortunately, it hasn't been too popular with candidates. However, in the gubernatorial election, the Green party candidate is running on a platform of being the only "clean candidate". Both the republican and democratic candidates are privately funded. As a result of the public funding, this candidate (who goes by the name of Jonathan Carter), has received far more media attention and had far more TV ads than other minor-party candidates have had in the past. Of course, it's still unlikely that he'll win, but he's definitely more than just an also-ran in this elections, which is good, I suppose.
Besides Maine, I know that Arizona and New York also have clean elections laws. I don't know of any other states off-hand, so message me if you do and would like to me to amend this list. Note that clean elections mostly takes place at the state level. It's unlikely that something like the presidential election will be de-privatized for a long, long time to come.