I'll be discussing the Authorization part and then we'll send a Tweet.
I'll be using the Tweetsharp assemblies which is available here. It supports Windows Phone 7, and will probably support Mango too, but keep an eye there.
If you don't know how OAuth for Twitter works, visit apiwiki.twitter.com. You will also require Consumer Key and Consumer Secret from dev.twitter.com for your App, sign in there with your Twitter account, register an desktop client app and you'll get the Consumer Key and Consumer Secret.
Now following will be the OAuth workflow of App using Twitter:
1. Get OAuth Request token.
2. Get the Authorization URL and redirect user to web browser and allow User to grant access.
3. Get back the 7 digit code via User.
4. Get OAuth Acces token using Request token and the 7 digit code.
5. Authenticate Twitter Service with Accest token.
Done!