The answer to 'my light fixture won't turn on' is that there could be several common causes, such as a burned-out bulb, a tripped circuit breaker, or a problem with the wiring or switch. Symptoms may include the fixture being completely dark or flickering/dimming.
To troubleshoot, first check the light bulb and replace it if necessary. Then check the circuit breaker to see if it has tripped, and reset it if so. If the problem persists, there may be an issue with the wiring or the light fixture itself, which would require the help of a professional electrician.