Part instrument and part looper/recorder, Thumbjam is well regarded in the community as one of the first and best iPhone music apps. Development has continued to keep pace with new technologies, as well as adding Universal iOS support.
As an instrument there are many samples available for playing with the on screen note bars. As the name implies, you can jam out fully with just your thumbs and a small learning curve. Importing your own samples is made easy with AudioCopy/Paste. There is even support for creating multi-sampled instruments, using a specific sample for different octave ranges.
The looper allows you to lay down quick melodies, or even import a backing track from another app. A lot of thought has gone into this app, but this is most evident in the looper where each recording is a separate instance. If you mess up, you can just keep hitting the "Delete Last" button to only remove the latest loop, instead of starting over from scratch.
On iPhones this is practically an Essential app, you really can jam with just your thumbs, but on larger iPad screens the experience is a little awkward to play.