It sounds like you have an electrical problem or a problem in the ignition switch. When you turn the key to start the ignition switch makes a connection with the starter solenoid or relay which will allow current from the battery to the starter. When the engine is started the ignition switch is returned to on from start and the connection to the solenoid (or relay) is broken, cutting power to the starter motor. The reason it works this way is that the current to the starter is boosted by a coil to high voltage, which would be dangerous to have running through a switch your hand would contact, using a solenoid or relay lets a small voltage control a big voltage.
Here's a diagram of a starter system:
As for what is going wrong it could one of the following:
- Faulty ignition switch: maybe someone tampered with it to try and run it without a key, or maybe it is stuck to start. Try twisting it to off if you haven't already. It's also possible it's simply broken and making contact no matter what you do
- Hotwiring: it's technically feasible to hotwire a car that doesn't have an immobilizer on it, of where the immobilizer has been disabled. An immobilizer is a system which will prevent the car from running unless a specific key or dongle is presented. Bypassing the ignition key mechanism to present power to the starter system
- Electrical fault: If the right wire gets its insulation worn off and comes in contact with power then the starter will run
- Solenoid/relay bypass: maybe someone took the switch out of the equation
- Malfunctioning solenoid or relay: the relay could be stuck closed, allowing power to the starter
- ID10T error: never dismiss the power of stupidity, someone might have deliberately run electrical connections or messed with something without understanding the possible results
I would start with inspecting the ignition switch for tampering or function. If wires are dangling all over and someone's used wire nuts or just left them bare then it's a good indication. Then I would inspect the starter motor and relay connections. Disconnect the ignition switch connections from the solenoid/relay and then connect the battery again, if the starter still engages then the solenoid/relay has been bypassed or has malfunctioned. If the starter doesn't engage then the relay is working and there's no bypass, so the problem is upstream. Use a multimeter to detect power coming down and work your way back to the ignition switch until you find the culprit.
Here's a good page on ignition systems and wiring.