How do I know what size grounding conductor is required?
If you are in the US, you'll likely follow National Electrical Code (NEC).
Lets start with some definitions.
NEC 2008

Bonding Jumper, Main. The connection between the grounded circuit conductor and the equipment grounding conductor at the service.
Grounding Conductor. A conductor used to connect equipment or the grounded circuit of a wiring system to a grounding electrode or
electrodes.
Grounding Conductor, Equipment (EGC). The conductive path installed to connect normally non–current carrying metal parts of
equipment together and to the system grounded conductor or to the
grounding electrode conductor, or both.
Grounding Electrode. A conducting object through which a direct connection to earth is established.
Grounding Electrode Conductor. A conductor used to connect the system grounded conductor or the equipment to a grounding electrode or
to a point on the grounding electrode system.

and a picture to help make things a bit more clear.

The size of the Main Bonding Jumper, Grounding Electrode Conductor, and Grounded Electrode is defined by table 250.66



The size of Equipment Grounding Conductors (the ones that run from the Ground bus bar out with each circuit), is defined by table 250.122.