You can login to the application to see how the API works from the perspective of a user, or you can view this video that walks through a sample scenario.

The process goes something like:

1. user clicks a link on your site to login
2. user is taken to twitter for login, if they are already logged in proceed to the next step
3. user is asked to allow your application access, they accept or deny (only happens once)
4. user is taken back to your site, where your site can ask twitter for information about the user (name, screen name, etc) and create a user-profile based on that.

Some helpful links I found:
First, a document on twitter’s site that describes their API and gives a technical overview of the login process.
Second, a document that provides php source code that you can test and read to learn about how to implement OAuth/Twitter login. Example in Php: using Twitter’s OAuth

I’ve done a short video series that follows me through the process. These were done “on the fly” as I worked through the problem. FYI – no source code will be provided for download, and please remember to review the terms of service for any site that you scrape.

For example, I tweeted “@calculatorbot define capricious” and calculatorbot responded “http://xr.com/d4es capricious: (2 defns) Adjective: determined by chance or impulse or whim rather than by necessity or reason”

Calculatorbot includes a url to the word’s definition and the first definition of the word in its reply. Also note that the url returned is a shortened url.

Sorry the details are scarce, but it will just have to be that way for a while.