The People application allows you to integrate incoming feeds from various sources. Mine is configured to get feeds from Twitter, Facebook, Skype and Windows Live. You can also use the same application to Tweet and post to Facebook. Since it is a Live Tile I set it to large format and see notifications through it.
If you want an application from Twitter itself, I guess it will be like wishing Facebook would make one. The Facebook application on Windows Phone 7 was written by Microsoft, and I know from the press releases that Microsoft won't do that again for Windows 8.