It's a measure of the minimum feature size that can be resolved by the lithography system. In non-engineering speak this means it's the smallest size (in one dimension) that something on the chip can be drawn and still actually be able to be created in the fabrication process. Typically this "smallest sized" object is the physical length of the polysilicon transistor gate. The actual transistor gate length is much smaller than this.
I'm not sure if I made this clear enough... write back if you want me to elaborate.
Patrick Mahoney
IA64 Microprocessor Design
Intel Corp.
* Not speaking for Intel Corp *
Edit: There's a little more to it than this. I quoted the textbook answer of what a process size is, but generally everything on a chip is shunk when the process is defined. So, all things being equal, a chip on a 0.18um is typically ~25% smaller than one on a 0.25um process. In addition to smaller transistors and smaller wires (and thus smaller chips), smaller processes should result in lower power designs which run cooler. Smaller processes also result in faster chips because essentially the smaller a CMOS transistor, the more current it is capable of driving.