Having a quick and easy method of instantly recording and replaying music is a valuable teaching tool which I sometimes use during piano lessons, to illuminate specific points. Sometimes, hearing particular issues, rather than discussing them, is a much more effective way of understanding what is happening, whether these are problems to be resolved, or elements worthy of congratulation!
I have two bits of equipment which I use to make recordings. One is my laptop, which will record very easily using its built-in microphone, with just a few clicks of the mouse. To make an even less obtrusive recording, my mp3 player also has a built-in micophone, and is small enough that I can start the recording without the student realising what I am up to. This is useful, if I think that knowing that they are being recorded will make it impossible for them to concentrate on playing the piece. I can also decicde if it would be a good or a bad idea for them to listen to themselves play after they have finished.
Neither method produces a quality recording, but it is usually enough to pinpoint the areas for action.
This article comes with a health warning; listening to yourself play is sometimes a very difficult and negative experience and can severely dent your confidence. It is important that analysing the recording becomes something that you can build upon, rather than a time for plummeting into the depths of despair!
Having said that, once the students become used to the concept of recording and analysing their playing, I will often encourage them to use this as a way of improving their work.