Well these problems can be difficult to diagnose.
There are three basic things to check
1. Fuel
2. Spark
3. Air
1. Fuel - When you first turn the key to the ON position (not start) you should hear the fuel pump in the gas tank (under drivers side rear seat) come on for about 2 seconds. If it sounds like it is gurgling, then it could be at fault. If it comes on, sounds healthy, shuts off, then you need to check for fuel pressure at the fuel rail (on the pass side of the engine, sharader valve looks like a bike tire fill up) Careful there is a bunch of pressure in there, press the vavle down briefly and see if it comes shooting out (use a rag so it doesnt go everywhere. this will tell you if you at least have fuel pressure at the engine
2. Spark. Pull the #1 spark plug wire and check for spark while the starter is being cranked. You said it runs for a second then shuts off, which probably means that your sprak is just fine.
3. Air/Vacuum. It is not uncommon for the early Explorer's (like yours) to have the lower intake gasket leak because the lower intake manifold bolts work themselves loose. This is a little more tricky to detect, so try the fuel stuff first then get back to us. (less typing)
Dont start throwing $$$$ at it, meaning dont go replacing parts yet, the fuel filter is cheap, and also a real pain ti replace, and it should be replaced every few years.
Yours is a 91 so it has a MAP sensor and a single oxygen sensor. These things can cause this situation, but we can worry about those after you eliminate the fuel system as the problem....
My guess is either the fuel pump, fuel filter, fuel pressure regulator, or fuel pump relay from what you describe....
Do you have a Chiltons manual?
Does it blow smoke when it's running?
Do you smell "rotten eggs"?
