Contactor was burned and wouldn't stay in on its own. We replaced it. No fix. So replaced wiring from the thermostat to the unit. Still no fix. If you depress the plunger on the contactor it WILL NOT stay in.As soon as you let go it pops right back out. Also, even if you hold it in the compressor makes a weak attempt at trying to start but it's a no go. However, if you hold the plunger down, the fan will run (until you let go and it pops back up)).

Sounds like you don`t have low voltage(24v),does the inside blower come on? check for broken tstat wire, do you have a mulit-meter to check for power?

You probably didn`t need a new contactor, the 24v is what make the contactor pull in and stay

Thank you for getting with me. It's really getting hot here. The inside blower does come on. We are trying to locate a meter to check for power. The contactor looked like it was in really bad shape. It had black burn marks all over it. But here is the thing. With the old contactor, we could hold the plunger down for a minute and it would stay in on it's own, this enabled both the outside fan and the compressor to run (until the thermostat turned them off). With the new contactor, it doesn't matter how long you hold it down, it immediately jumps right back out. But, while holding the plunger down, you can hear the compressors very weak attempt to start and then it stops, But the outside fan sounds just fine (until you take your hold off the plunger of course)so, if it isn't getting 24v, and it isn't thermostat wire, then what could it be?

And... where and what is the tstat wire?

Sorry,tstat is short for thermostat, the small wire that runs from the furnace to the contactor is a thermostat wire, the larger wire is high voltage