I've been looking around for a really simple API that would be a nice place to get started using web services from PHP - and I realised that bit.ly actually fits the bill really well. They have straightforward api docs on google code, and it's also a pretty simple function!
Here's a simple example, using PHP's curl extension, of using the bit.ly API to get a short URL, using PHP (you need an API key, but if you're a registered bit.ly user, you can log in and then find yours at http://bitly.com/a/your_api_key).
$ch = curl_init('http://api.bitly.com/v3/shorten?login=username&apiKey=R_secret&longUrl=http%3A%2F%2Fbeta.lornajane.net');
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
$result = curl_exec($ch);
print_r(json_decode($result));
The API also provides methods to get the long URL from a short one, validate a short one, and also get all sorts of statistics about a URL that you've shortened using bit.ly. I'm increasingly using bit.ly for shortening URLs, and also for their awesome bundles feature where you can give one link to refer to a list of resources. This comes in really handy for example when I'm giving talks, I can just refer everyone to one link to find everything that I mentioned! Using the API for these kinds of sites mean that we can integrate with systems that already talk to bit.ly, and also avoids us re-inventing a similar service. There are other ways to do this from PHP but since curl is a core extension, this should work on pretty much all installs.
* This is my personal blog, I can use function names as verbs here if I want to