Try turning the headlights on, and watching their brightness while you try to start the motor. If the headlights dim way down or go out when you try to start, then the battery is either too weak to run the starter; or the battery cable connections are corroded or dirty. (Corroded battery cable connections often look just fine to the eye; so don't dismiss them as a possibility because they look clean. Use a tapered reamer type battery cable service tool to scrape a shiny edge on both the inside of the cable clamps, and the outside surface of the battery posts.) If the lights go dim or out, and cleaning the cable conections does not help; then the battery should be charged with a battery charger for at least 10 hours. If the battery starts the car after being recharged; then it had either become discharged by a light or electrical accessory that was left on, or by a long period of storage, or by a defective alternator. After charging the battery; if the lights still dim when trying to start, the battery will have to be replaced.
If the lights do not dim when trying to start; then either the neutral safety switch (only used on automatic transmission vehicles) or the clutch pedal position switch (only used on manual transmission vehicles), or the starter relay is probably bad. A defective starter motor could also cause this problem; but it would be far less likely than one of the above causes. If your car has an automatic transmission; try moving the shift lever to Neutral and see if it starts in that position. Sometimes a neutral safety switch will stop working in Park, but will still work in Neutral. The clutch pedal position switch will sometimes not permit the starter to run if a thick carpet is placed on the floor under the clutch pedal (which prevents the clutch pedal from being pressed all the way down). Using a jumper wire to run battery power to the starter solenoid will bypass the starter relay and safety switches, and should make the starter run (unless the starter motor is bad). This should only be done by a qualified person; as on some vehicles, the wire from the ignition switch must be first disconnected from the starter solenoid in order to prevent damage to the ignition switch.
A defective starter relay will act just like a bad neutral safety switch or clutch pedal position switch. There is a big difference in the price of aftermarket starter relays. Airtex/Wells has one of the lowest priced relays (which is less than 1/3 of the price of some other brands). At about $20; it would be worth installing one, just on the likelihood that the original is defective.