Sounds like an aftermarket security system. The factory system would have an anti-theft light in the dash (and was only availiable in the limited in 93-94 IIRC) There should be some kind of box that controls it under the dash, it will probably be fairly obvious and will have the name of the manufacturer on it. You can also look at the back of the keyfob if you have one.
You probably have to arm/disarm it to get it to stop disabling the starter. This can be a problem if you do not have any remotes for the system. Sometimes depending on the alarm there is a small pushbutton somewhere which can be held or pressed in a certain sequence (along withe turning the key on/off possibly) to disarm the system.
If you can't get it to disarm, you can also find the starter disable relay and bypass it so it will start. It's probably connected to the ignition switch wiring which comes off the top of the column. Having a alarm system with no remotes just hiding out in the vehicle is a recipe for problems, especially if you have the battery go dead.
Many times after the battery is dead/disconnected they go into a no start mode where you need to arm/disarm them to prove you are not trying to trick the alarm and steal the car by messing with the battery. This is a problem if you can't do that because there is no remote.
I spent 20min messing with a Mazda MPV one time after jumping it... the guy didn't even know it had an alarm. By chance I found a pushbutton switch under the dash and some random combination of pushing it and turning the key on and off disabled the alarm and then it was able to start.
