Equipment Grounding Conductor
Equipment ground is the wire that grounds metal parts to prevent them from producing a shock- by allowing a circuit breaker to trip when a faulty wire touches metal parts and current flows through the ground wire. Because the tripping occurs instantly, the equipment ground wire will not heat up and can thus be smaller than the hot wire. Table 250.122 in the NEC shows the ground wire sizes for a given size of a circuit overcurrent device:
The Grounding Electrode Conductor is the wire than connects frin the main breaker box to the ground rod. Its purpose is to even the potential between the earth and the main breaker box. It does not function to trip breakers. See BS Alert at Ground Rod Resistance
The above is not the same as the grouned conductor - see explanation at . The grounded conductor carries current - but it is grounded as well. There appears to be controversy whether you get shocked if you touch the grounded conductor - because it is at 'zero' potential. The answer is yes - under the condition that there is a potential difference between the grounded conductor and whatever you are touching (unless you are floating while touching the grounded conductor). This also explains why power line workers can hang from a helicopter and touch live transmission lines : it is safe as long as you don't create a current path through your body. Air is a good insulator, so that hanging in air can evergize you to many kV while not getting shocked.