Conweets is a useful web service that can be used to track conversations between two twitter accounts.
The service has now launched their official mobile app for Apple iOS devices.
The mobile app provides the same feature in an optimized visual layout. You can view conversations between two different accounts. Or multiple conversations started by a single account.
The most useful scenario is to locate your own conversations on Twitter from the past as Twitter does not make it easy from their own user interface.
The app is available as a free download.