According to Twitter, the bug has been corrected and it has seen "no indication of breach or misuse by anyone" after an investigation.
The bug itself is related to the hashing function Twitter uses to mask passwords. Twitter says passwords were written to an internal log before the hashing process was completed, leaving them exposed. From Twitter:
We mask passwords through a process called hashing using a function known as bcrypt, which replaces the actual password with a random set of numbers and letters that are stored in Twitter's system. This allows our systems to validate your account credentials without revealing your password. This is an industry standard.
Due to a bug, passwords were written to an internal log before completing the hashing process. We found this error ourselves, removed the passwords, and are implementing plans to prevent this bug from happening again.
Out of caution, Twitter users should reset their password for the service, as well as those for any services using the same password. Now would also be a good time to start using two-factor authentication if you aren't already.