You'd probably want to have some kind of switch that only turns on the LED when the glove compartment is open (or when you manually turn it on.) So, 12v (from the fuse panel) to the switch, to the light source, to ground.
For the light source, you can either use an automotive 12v LED strip, or wire in a standard socket for a festoon or other standard bulb type (such as a 194.) A socket would be more work to install cleanly, but you'd be able to easily swap bulbs (LED, incandescent, color, whatever.) A LED strip would probably be much easier to install (just stick it in place) and could generate more light over a greater area, but would be much more difficult to replace if the strip becomes damaged.