Google's URL Shortener has been used by Google services like FeedBurner, Google News, Blogger to share links on Twitter. Google didn't provide a web interface for the service, but third-party sites managed to fill the void.
Now you can go to goo.gl, paste any web address and get a short URL. Google also shows stats for any short URL generated using the service: append ".info" to the address and you'll see the number of clicks, a list of traffic sources and some aggregate information about visitors. Here's an example: http://goo.gl/l6MS.info.
If you sign in to a Google account, Google will save a list of recently generated URLs. Another advantage is that Google will always generate unique URLs when you are logged in, so that the stats are more useful.
Google's URL shortener still doesn't have an official API and it doesn't offer all the features that are available at bit.ly (most notably, you can't customize URLs), but it works well. "We've had near 100% uptime since our initial launch, and we've worked behind the scenes to make goo.gl even stabler and more robust," mentions Google's social blog. Google also added automatic spam detection and doubled the service's speed.
If you use Google Chrome, install goo.gl URL Shortener, an unofficial extension that generates short URLs and copies them to the clipboard. For Firefox, there's a similar extension called goo.gl lite.