We are as worried as you are, we try to find what’s going on with our app. And I think we’ve found it. Let me show you some pictures first:
1. twitter’s official app:

You see, it won’t let you log in.
2. echofon

no, it won’t let you log in.
3. twitterific

no, it won’t.
The other 3 twitter apps on iPhone won’t let you log in as well as twitbird. Because it’s not us who don’t let you log in, but the twitter itself, it has something wrong. Please see this page:
http://j.mp/9Z1lnL
You can see the first update saying Oauth is affected.
What is oauth? oauth is some tech stuff, it let you log in to your account safely.
We are sorry, and we are contacting twitter. This is not our fault, it’s bad to you, it’s bad to us. We are trying everything we can to fix the issue.
If you have any problem, tell us, we will try to tell you how’s the progress.
We are sorry. We have moved the apps off the shelf, and we will make it available again when twitter solved the issue.