In a recent post Hasin Hayder has taken a look at using the OAuth PECL extension (this one I assume) to connect your application's login system with Twitter's authentication backend.

if you are interested in developing twitter applications, you must have read about twitter API and it’s authentication protocol. your application can fetch user’s private data but it has to authenticate itself as the user for that. so there are two ways to do it: asking user to provide his twitter username and password [...] or let twitter handle the authentication on behalf of you.

This second option is where OAuth comes in. Once you've registered your application on Twitter, you can create a token and send it over to their site for validation. The idea is that, since the user has already authenticated on the Twitter site, they can allow an external application to "share" that login information/process and let the remote application fetch information about the user from he Twitter API.