Unobtrusive AJAX Star Rating Bar is a rating system made in PHP by Ryan Masuga. It is easy to install and intergrate into any PHP programs. One easy function -- rating_bar() can do everything you need with this script.
Because it uses Ajax, post a rating does not need to refresh the page. It got a built-in anti-cheat system(IP). Here is qoute from the official site

This is a rating bar script done with PHP and mySQL that allows users to rate things like can be done on Netflix or Amazon, all web 2.0-like with no page refresh. It is a major improvement on the previous version because it is now unobtrusive, meaning that if Javascript is off it will still work (although the page will refresh). You can also set the number of rating units you want to use (i.e. 4 stars, 5 stars, or 10 stars) on a rater to rater basis (see samples below or read the docs). A few other changes were made as well - see the docs for details. Note that this script isn't tied to any specific system (such as WordPress), so you should be able to adapt it to your situation without too much trouble.

Editor Comment:

The best simple rating system yet! Give it a try.