If you go out and hunt for a DJ app that lets you mix together two different tracks, you're going to come across a lot of really, really, bad design. Even the best stuff is a hell of fake spinning turntable skeuomorphs. Native Instruments' Traktor DJ for iPad is completely different.
Traktor DJ shows you the two tracks you're mixing together the way they exist in the digital world: As linear waveforms. The app finds the beats of your music using the same processing algorithms as Native Instruments' well-respected Traktor computer software. Once you press play, cueing up the next track, setting loops and syncing tracks from the songs you're playing is simple. The loop function in particular is dope: use two fingers to select the part of the song you want to loop, and it'll pull that chunk in real-time with zero interruption.
The app's tap based interface has lots of little smart little touches that makes it feel like the app is working with you rather than the other way around. I've got a lot of experience with music, but I'm not a DJ at all, and I found the app to be an incredibly fast learn. It's intuitive enough that I could see people with no music experience picking it up and fooling around on the couch or on the bus. For example, I have no idea what music to mix together, and luckily the app analysed my iTunes library and suggested some matches. And there's plenty of advanced features for those with the know-how.