I think we all memorise in different ways, depending on our 'learning style'. Some of us remember better visually, some are better at retaining the information audially (aurally?). Some do a combination of both.
With my jazz band when we were working out a new tune, I always used to listen to the original a few times and then memorise it aurally. In that way I built up a mental library of probably well over 100 tunes, all stored in my head - no problem.
Now, however, we're doing a lot of writing ourselves and at most rehearsals I'm given a sheet with a new tune plus arrangement to learn. Not wanting to use the sheet music on stage, I found I was learning some of it by visualising the notation, but then found a method that worked better for me, using the Edirol - recording rehearsals and then listening to the new tunes over and over again. It works and is a great reason for investing in some home recording equipment!
For some unknown reason, all of us in our band find that once we've performed a piece live, it mysteriously 'embeds itself' and becomes a permament part of your mental repertoire and even if you'd struggled with it in the past it now becomes perfectly easy to remember. I have no idea why this happens, but it's certainly another very good way to memorise a piece - perhaps the best, in my experience.
I can still remember note for note a piece I played at a music festival when I was 12, even though I played it from the sheet music at the time. Weird!
I would imagine it's harder for pianists to memorise than single-line players for the simple fact that you have so much more detail to remember, but because I'm a hopeless sightreader on piano and once wanted to learn a few nice piano pieces I once made myself memorise the whole of Debussy's 'Clair de Lune', studying about 4-8 bars a day and learning them by heart as I went along.
I do think that if we can hear any tune in our head then we must have the ability to memorise. If what we're trying to memorise doesn't really resemble a tune, then perhaps the best way to proceed is to do it bar by bar, using whatever techniques work best for us - using intervals (as described by Tsax), audially or visually or a combination of all three.
I also think you can play with more expression once a piece is fully committed to memory, though I know others will disagree....
I do believe that the ability to memorise is a very important musical skill to develop; when you think back, the notation of music is a relatively recent development in the history of worldwide music-making. I went to a Greek violin workshop a while back which was very interesting and illuminating. This old Greek man taught a whole group of us a long complex piece without any notation at all, using the traditional method of taking it phrase by phrase from the beginning using listening followed by repetition and constant recapitulation - the method worked because we virtually all managed to learn the whole piece by the end of the session, even those of us who'd never memorised anything before.
Violinia