To use this feature, you'll need a Google account. If you don't already have one, simply create one.
Your profile will keep a list of all the results you +1, but you can delete a recommendation if you no longer find it useful.
This feature will be rolled out slowly, but if you're eager to start +1'ing, you can opt-in now by visiting Google's experimental search site.
They're also planning on providing a plug-in for third party websites so you can +1 a site after you read the content.
So far, this will not reorder Google search results, but you can see how a link's credibility goes way up if someone you know and trust +1's it.