Take a look at the Zend_Oauth example, which shows how to login to Twitter using Oauth: http://framework.zend.com/manual/en/zend.oauth.introduction.html

You can read up on the details on various Oauth web sites, but in a nutshell, Oauth means that you request a "request token" from Twitter using PHP, and then redirect a user to twitter along with that request token. On Twitter, the user will be asked to grant your application access (via this request token), then they'll be returned to your site. On your site you now try to exchange the "request token" for an "access token". This access token is then used to interact with Twitter on that user's behalf. You should store it in a database and associate it with that user. Unless that user specifically removes the permission from your application via their control panel in Twitter, you can continue to interact with Twitter on their behalf.