Buzz buttons are used for starting conversations about interesting web content ("Hey guys, what do you think about this news story?"). +1 buttons recommend web content to people in the context of search results ("Peng +1’d this page"), and +1's from social connections can help improve the relevance of the results you see in Google Search. You can use the +1 button, or the Buzz button, or both—pick what’s right for your content.
Personally, I still think there will be a lot of confusion between Google Buzz and Google +1. When you "like" stories in Google Reader, should that be a +1 but when you comment in Google Reader, should that be a Google Buzz? I don't know.
Matt McGee at Search Engine Land has two good stories on the button. The stories are named:
How will it affect SEO and results? Google said:
Content recommended by friends and acquaintances is often more relevant than content from strangers. For example, a movie review from an expert is useful, but a movie review from a friend who shares your tastes can be even better. Because of this, +1's from friends and contacts can be a useful signal to Google when determining the relevance of your page to a user's query. This is just one of many signals Google may use to determine a page's relevance and ranking, and we're constantly tweaking and improving our algorithm to improve overall search quality. For +1's, as with any new ranking signal, we'll be starting carefully and learning how those signals affect search quality.
When will it show in analytics? We do not know.
The one thing not mentioned in most blog posts is the code. Google is actually recommending using the HTML5 syntax instead of what Google gives you on the button creation page. The code they are recommending is: <div class="g-plusone" data-size="standard" data-count="true"></div>
You can find out more details and stories on this at Techmeme.