There's a special spring-loaded cog at the end of the starter that's meant to engage when the starter is going and to retract when it isn't. If this special(Bendix) cog don't retract, the starter will remain engaged. You'll have to pull the starter and look at the axle at the end by the flywheel.
If you're implying that it engages without you turning the key, the problem is one of the following: * a faulty wire somewhere, * the starter solenoid was connected wrong * or the solenoid return spring has broken.
If you can hear your starter motor turning but not the engine you have some teeth missing on the ring gear. If you can hear nothing, not even the starter motor solenoid clicking into place, you have a connection problem between ignition switch and starter motor.
Check you connections make sure there on the correct way.
You should locate and test the solenoid. Most have three connections. There is a large cable from the battery to the solenoid, from the solenoid to the starter and from the start switch to the solenoid. The power terminals to the battery and the starter are large and have large cables. The starter switch connection is much smaller. Turning the key to 'start' provides a low current voltage to the solenoid causing the solenoid relay to close and power from the battery cable to connect through to the starter. If that voltage is present then the solenoid or the wiring between the solenoid and battery or solenoid and starter is suspect. If that voltage is missing then the starter switch or solenoid feed line may have failed.
This usually means a sticking solenoid that keeps the starter engaged after activation. The fix is to remove the starter and clean but my s-10 starter is a closed unit so you will have to replace the complete starter and solenoid unit.
You could have a sticking solenoid on the starter, you could have a defective starter button, or your solenoid wires may be set up in a way that causes the current to ark between points when it's not supposed to.
The soleonoid on the starter is bad. When you buy a new starter it comes with the solenoid. Try tapping on the starter with a hammer if it starts after that then this is your problem
Did you put the small solenoid wire on the correct small terminal?
there is no relay for the starter. there is a solenoid (switch) and ignition switch. the solenoid is part of the starter (small cylindrical shape above starter housing. the solenoid has the battery and ignition connections). and the ignition switch is where the car-key goes. you need at least 8v from the ignition to the solenoid for the starter to operate. use a multimeter. otherwise check your starter - battery connections. otherwise replace the starter. otherwise, take off the starter and inspect inside the starter housing. if the starter is clicking but not turning the engine the starter teeth aren't engaging with the teeth on the flywheel. get a manual to the car here: http://www.germanautoparts.com/Books/Volkswagen
ckeck the solenoid on the passanger side front fender
Bad solenoid
Starter solenoid is bad. I assume you mean engine turning over, not starting, but continuing to turn over after key is released. Disconnect battery cable from post, then replace starter solenoid.