If you have a Google account (and are logged into it), then pressing this button for sites, articles and search results you like will actually change the search results people in your online social circle see when doing searches in Google. In other words, if a site you clicked on shows up in the search results of someone you know, it may then show up further up in the search results since Google knows that a friend (you) of the person searching liked that result. For those of you familiar with Digg or Liking on Facebook, this is a very similar concept, except it is fully integrated into Google which makes it very powerful (if you use Google).
This is still experimental, and who knows if Google will keep this feature in the long-run, but if you want give it a try you might already see it in your Google search results (it is being rolled out now to users of English-language search).
You can also try the button here:
which will add a +1 to this post, and/or you can press the button in the upper right of this page to +1 this site as a whole.
Of course, if you're reading news posts in the future, or are looking at older posts on this site that you like, please press the +1 button at the end of each post to help increase its visibility on Google and I appreciate all the help I can get, so +1 away!
If you start using this feature here and on other genealogy sites around the web, this should increase the relevance of the search results you and your friends get online as more and more of your social network starts to contribute information (instead of it just being determined by a computer algorithm). I don't know yet if it will also change your own search results based on the sites you click on, but I guess that doesn't matter much since Google already knows about which sites people are visiting. Perhaps pressing +1 will rank more important than visiting the site (since you may have visited a site and not liked it).
For those interested in adding this button to your own web pages, Google has a page that will generate the code for you automatically. One thing they leave out, however, is that the link they give you will increase the count of only the page it is on, not of the site overall. For example, when this article is posted you will see it on the main page of this web site, which is different from the permanent link to this article, which is also different from the page you would see if you searched for all articles with the 'google' label. If you take the code Google gives you (from the previous link) and place it on your web site (like in the top right of the page like on this web site) it will actually give a different count for each of those views since the web address is different in each case.
Originally I figured this out by looking at how Blogger (which this site runs on) was adding the +1 buttons to posts, but it turns out the ability to direct a +1 button to a specific web address is fully documented on the +1 button API page. For example, the +1 button in the top right of the page has a specified web address for the site's home page to insure the count is for the site as a whole and not whatever page the person happens to be on at the time. Indeed each button that Google automatically adds to the bottom of each Blogger post (like on this site) links to that specific post. That's why you can have multiple versions of the button on this page and each one is specific to the post and not to the page (which could contain multiple articles).
p.s. You'd think Google would be smart enough to come up with a name that could easily be converted into a verb. People Like things on Facebook all the time (Like in the sense clicking on the Like button for a page, post or comment) and they Digg posts on sites that support Digg. When I wrote "+1 away!" above it felt a bit contrived. +1 to Facebook and Digg for coming up with the better terms...