- RepRap is short for Replicating Rapid-prototyper. It is the practical self-copying 3D printer shown on the right - a self-replicating machine. This 3D printer builds the parts up in layers of plastic. This technology already exists, but the cheapest commercial machine would cost you about €30,000. And it isn't even designed so that it can make itself. So what the RepRap team are doing is to develop and to give away the designs for a much cheaper machine with the novel capability of being able to self-copy (material costs are about €500).
The RepRap project is a bit optimistic in their description -- their "self-replicating" machine is capable of printing all of the non-commodity parts of their machine, but still needs stepper motors, steel rods, screws, nuts, control circuitry, and of course a human to assemble the pieces.
- http://www.reprap.org/bin/view/Main/McWire_Cartesian_Bot_1_2 - easy to follow instructions for a kit version of reprap.
- decide what system to build
- what print materials? ABS, PCL aka Shapelock, PLA, ...
- multiple print heads?
- what chassis design? (lasercut acrylic, plywood + steel rod, ...)
- control electronics design (something better than "one Arduino to rule them all" please)
- purchase parts
- build it!
Some parts of RepRap can be developed independently. In particular, the Extruder "print head" is pretty much independent of the chassis mechanism, and looks like a fun project. Might want to consider other print mechanisms too; this one may be specific to a few of the thermoplastics.
The Builders wiki has some very useful content.