Unfortunately this is not available on the BlackBerry yet. The app your are referring to is called Shazam and is making some waves on the iPhone ATM.
Some other forum posters have suggested the following:
AT&T customers have the Music ID service AT&T - Music is a place to find all your mobile music needs.
In the UK if you call 2580 (IIRC) from your phone and hold it up to the music Shazam will text you back with the track.
Virgin Mobile also has a similar service although from what I am reading you get charged some pretty high fees for all of these solutions.