Why do I have a green light on my GFCI but the outlet will not come on?
Use a voltage tester like a "Wiggy" or a neon tester to test the presence of voltage at the wires.  (not just in the socket of the GFI)  If there is current in the wires, double-check that you have connected the power to the line side of the GFI, not the load side.  Be sure that the neutral (white wire) and hot (black) are connected to the correct terminals and not reversed.

Remember: GFIs are designed to protect from ground faults (where current is being shunted to ground).  They are not circuit breakers and they do not protect against over load of current like a fuse would.

If there is no power at the wires then the problem it outside the scope of this question and you may wish to hire an electrician.