My blankety blank gauge doesn't work!
All electrical gauges in your car (Gas Level, Amperage, Oil Pressure, Water Temperature) are just volt meters. The power comes out of the fuse box to the gauge where it goes to the "sender" that is actually a receiver. The voltage goes through the "sending" unit to ground and the amount of voltage drop is what is displayed by the gauge.
The gauges work on a nominal 12 volt system so any resistance in the circuit will affect the reading on the gauge. In the case of a gas gauge it should read from 0 to 90 OHM resistance from the line going into the sender to where it comes out again (the ground wire). If that works you know you have a good sender.
If the gauge moves from when disconnected (isolated by near infinite resistance to a 12 volt system by a foot or so of air gap), and then moves when grounded you know the gauge works.
If you can read voltage in the feed wire to the guage and it reads the same at the back where it enters the sending unit then your wiring is ok; and the only thing left is a bad ground.
Somewhere in your car a bit of rust has built up under some part that is supposed to be part of the grounding circuit. This works for aftermarket gauges (which there are only gas gauges used in most brands of cars; either a 30 Ohm resistor used in older cars, and the 90 Ohm resistor used for cars made after .... I can not remember when.) But according to the Oracle of Google it was 1965 that GM switched over.