No start condition
It sounds to me like a dead battery. You need to perform a load test on the battery. you may have 12 volts comming out but not enough amperage to turn the starteer. if the battery terminals were dirty chances are the battery was not getting a charge from the alternator. Charge the battery or buy a new one. the clicking sound you heard was the starter solonoid clicking in and out because of the dead battery. Now that you cleaned the terminals, you have SOME more current flowing through the circuit but not enough to power the starter to start the car. either charge the battery or if its old, buy a new one.