Turn on the headlights, and have someone watch them while you turn the key to the start position. If the headlights go out or get very dim when you turn the key; then either the battery does not have enough power in it to run the starter (from age or from becoming discharged), or else the inside surface of the battery cable clamps and the outer surface of the battery posts need to be cleaned with a battery cable service tool.
If the headlights do not dim when the key is turned to the start position; then the neutral safety switch (on automatic transmission equipped vehicles) or the clutch switch (on manual transmission equipped vehicles) is either defective or out of adjustment. If your car has an automatic transmission; try moving the shifter to neutral, and see if it starts in that position. If your car has a manual transmission; be sure the clutch pedal is pressed all the way down when you try to start the motor. Sometimes, a floor mat or rug under the clutch pedal can prevent the pedal from moving down far enough to activate the clutch switch.
It is also possible that a fuse in the engine compartment has blown; or either the starter, the starter solenoid, or the igntion switch is defective; but those are less likely than the above causes.