[Joeaksa]

How about doing it right from the beginning? Dme relay should not stop it from turning over but will stop it from starting if its turning over. If it starts when the car is push started, then you have basic power to the car (fuel pump, injection and ignition is working or it would not have started when pushed) and something else is the culprit.

First check the battery and connections. Should be clean and tight. Then turn the headlights on and see if they are nice and bright. If they are not then you are not getting full power to the car.

If you have a volt ohm meter (VOM) then check the power at the battery with everything off, then with the ignition switch on, and then when you put the key in the "start" position. Get back to us with the readings. If you do not have one, you can get a cheapie one for $20 or so, and make sure its digital. Very good to have in the toolkit.

If everything is good so far then go back and check the 14 pin connector in the left back side of the engine compartment. I believe that the starter solenoid trigger wire goes through this? (someone correct me if this is not right)

If its ok then go to the starter itself and make sure that the solenoid wire has not fallen off.

As John said, starter could be getting hot and not working. How many miles on the car? If a lot then the starter could have seen better days.

After checking all of that you should have some more answers to what is going on. Next time instead of saying "it does not start" it might be better to say "it does not turn over" or "it turns over and does not fire" and so on. Gives a lot more information and helps us help you.