Sounds vaguely like the symptoms of a weak battery.
When you turn the car to "Start", it engages a relay/starter solenoid that connects the required high current directly from the battery to the starter motor. If the battery is weak, it has enough power to engage the relay, but when the starter motor attempts to pull high current to turn the engine, the battery can't handle the load, and effectively releases the relay. Because the relay is now open, the starter is no longer drawing current, so there is enough power (again) to engage the relay, then the whole cycle repeats. The result is a buzzing sound as the relay alternately closes and opens. (This is roughly how mechanical buzzers are designed).
Another symptom of this would be the interior lights dimming appreciably or turning off altogether when you attempt to start the car.
Sometimes, there is just enough impulse from the initial surge to turn the engine enough for it to start. Once that happens, the engine is running off of the alternator instead of the battery.