Google +1 is yet another attempt to make Google more social. It's Google's version of the Facebook "likes", a simple feature that's very powerful because it's part of a social network.
Google will show +1 buttons next to all search results and ads, while encouraging other sites to include the buttons. All +1's are public and they're tied to Google Profiles. The goal is to use this data to personalize search results and ads by recommending sites +1'd by your friends. Google Social Search already does this, but there's no support for Facebook likes, so Google had to come up with a substitute.
"+1 is the digital shorthand for 'this is pretty cool.' To recommend something, all you have to do is click +1 on a webpage or ad you find useful. These +1's will then start appearing in Google's search results," explains Google.
This feature is slowly rolled out to Google.com, but you can try it by enabling the +1 search experiment.