Services that consume or produce Twitter updates are popular apps these days, and there are more than a few on App Engine, too. Twitter provide an extensive API, which provides most of the features you might want to access.
Broadly, Twitter's API is divided into two distinct parts: The streaming API, and everything else. The streaming API is their recommended way to consume large volumes of updates in real-time; unfortunately, for a couple of reasons, using it on App Engine is not practical at the moment. The rest of their API, however, is well suited to use via App Engine, and covers things such as retrieving users' timelines, mentions, retweets, etc, sending new status updates (and deleting them, and retweeting them), and getting user information.
Most of Twitter's API calls require authentication. Currently, Twitter support two different authentication methods: Basic, and OAuth. Basic authentication, as the name suggests uses HTTP Basic authentication, which requires prompting the user for their username and password. We won't be using this, since it's deprecated, and asking users for their credentials is a bad idea. The OAuth API makes it possible to call Twitter APIs on behalf of a user ...