Ground-fault circuit interruption (GFCI) receptacles, are not wired the same as regular duplex receptacles. In a standard duplex receptacle, both receptacles and all terminals are directly connected together (Unless modified). If one half of the receptacle is powered, then the other half is as well. With a GFCI receptacle there are LINE side terminals and LOAD side terminals, which are separated by an internal switching mechanism.

The wires feeding the circuit are connected to the LINE terminals, which supplies power to the device. If everything is wired correctly, there are not ground-faults, and the device is not tripped, then electricity is allowed to flow to the receptacles on the device and to the LOAD terminals. So if everything is functioning as it should, there should be be power at both the LINE and LOAD terminals. However, if the GFCI device is tripped, there will only be power at the LINE terminals.

Resetting the GFCI

If the GFCI has tripped, it can usually be reset simply by pressing the RESET button. If you press the RESET button and don't feel/hear a click and/or the button doesn't stay in, it means there is a problem and the internal mechanism is not allowing the GFCI to be reset. You can try pressing the TEST button, then pressing the RESET button again making sure you press the RESET button all the way in. If the device still will not reset, you'll have to try and determine the reason.

Why won't a GFCI device reset?

There are three reasons a GFCI device will not reset.

Wiring is wrong

If the GFCI device is not wired properly (LINE and LOAD reversed, hot and neutral reversed, etc.), the device will not allow a reset.

There is a ground-fault

Obviously, if there is a ground fault, the device will trip as soon as you try to reset it.

There is a problem with the device

If the device has gone bad, it will (should) not reset. Some devices will continue to hold, even if there is something wrong internally. However once they trip, they cannot be reset. Other devices will trip as soon as something internal dies, and will not reset. This is why monthly testing is suggested. If you press the TEST button, and then are unable to reset the device. You'll be made aware of a problem sooner, and can have it repaired (hopefully) before any damage is done.

Rewiring a new device

Before you begin, turn off the power at the fuse/breaker box and make sure it's off.

Locate the supply wire pair

There should be an ungrounded (hot) and grounded (neutral) conductor pair (likely as part of a cable assembly), that supplies power to the circuit. As it sounds like you've already located these, I won't go into detail as to how to locate them here (there are many other answers on the site that explain this procedure).

Terminate the supply wire pair

• Connect the bare/green grounding conductor to the green grounding screw on the receptacle (and to the box if required), and to any other bare/green grounding conductors.
• Connect the (white) grounded (neutral) conductor to the silver colored screw terminal labeled LINE on the device.
• Connect the (black) ungrounded (hot) conductor to the brass colored screw terminal labeled LINE on the device.

Terminate load side wires

If there are devices downstream that require GFCI protection, you'll have to connect the wires feeding those devices to the LOAD side terminals on the device.

• Connect the bare/green grounding conductor to the grounding conductors in the box.
• Connect the (white) grounded (neutral) conductor to the silver screw terminal labeled LOAD on the device.
• Connect the (black) ungrounded (hot) conductor to the brass screw terminal labeled LOAD on the device.

Set the device

Once all the wires are connected, install the device in the box using the mounting screws. Install the face plate, and turn the power back on.

1. Press the RESET button.
   • You should have power to both receptacles, and any downstream devices.
2. Press the TEST button.
   • You should no longer have power at the receptacles, or any downstream devices.
3. Press the RESET button again.
   
   • Power should be restored to the receptacles, and downstream devices.

The most likely failure point here is the GFCI's. #2 is the switch. Then the wire-nuts. Lastly, the wire.

Since the switch is not involved, I'd try temporarily replacing the GFCI outlet with a quality plain outlet. Buy quality, preferably one which uses the screws to clamp down on the wire, similar to the way it appears your GFCI grabs the wires. Avoid plain backstabs: they don't use the screw and are cheap and should not be used twice even if you wrestle the wire out of them.

Obviously if changing the outlet fixes it, it's the GFCI. I wonder why (or if) you have two GFCI outlets daisy chained with the LOAD of one connected to the LINE of the next. It won't hurt them, it's just redundant, and so, a waste of money. It can also create confusion: if the outlet goes out and you go to the GFCI and it's not tripped, you'll go nuts trying to figure that out... until you discover the other GFCI that has tripped.

I would also get some wire nuts, they have improved much mechanically in recent years, they are also color coded. Use red ones to join 4 wires of that size.