PICAXE is a brand of micro-controller, running inside PIC micro-controllers from Microchip. These chips come in many flavors and sizes, from 8 pins to 40. They are very cheap and come with free software for programming in BASIC (MAC, LINUX, Windows) or with flow charts (Windows).
PICAXE is a nice alternative to other micro-controllers used in education in that it's cheap if you already have an electronics lab, the software is free.
I have used PICAXE to make data-loggers, for light, temperature, and wind speed. I have also made a MIDI controller, where the PICAXE interprets potentiometer metrics into MIDI controller signals; this is useful for running TRAKTOR, see MIDI and DJ mixers.
The main web site for PICAXE is here: http://www.rev-ed.co.uk/picaxe/
The PICAXE is programmed via serial communications from a computer. Note that zero volts is logic 1 and five volts logic 0, so the serial interface will need to be adjusted accordingly. Computer serial ports -9V logic 1 and +9V logic 0. See http://forum.sparkfun.com/viewtopic.php?f=14&t=9560 for a discussion of this problem, a solution being to program an FTDI chip to use the correct logic levels. A run-through of the solution can be found here: http://www.picaxeforum.co.uk/showthread.php?t=13589
One issue with programming is that many of the timing-dependent commands depend on the frequency of the chip, which changes between models and can be changed with the setfreq command. So, porting programs requires care to check and edit timing-dependent commands. Even the serial speed for programming depends on the chip.
As of 2010, the X2 series of PICAXE provides plenty of versatility and features. The 20X2 I have can operate from 1.8V to 5.5V, making it compatible with many different types and sizes of batteries. The 08M lacks much of the newer versatilities, but it is very small, 8 pins, and very cheap.