The differential is part of the drive train, located between the transmission and the wheels. The power from the transmission is delivered to the wheels through the differential. The differential is also the part that allows your wheels to turn and still be connected to the drive train. For instance, you turn hard right. Your right wheel will rotate much slower than the left wheel which has to move much faster to compensate for the turn.
Sometimes you will hear grinding noises while driving or turning and may experience some drag. The indicators a garage will typically find are metal shavings in the differential oil.
The rebuilt differential is not going to be cheap as you have found already. Plus the labor will likely be expensive as well unless you are comfortable doing yourself or have a friend or family member that would do it. I would either drive it until it fails completely, or trade it in for another vehicle. If it fails completely or locks up, you will not be going anywhere and will be stuck where you are at that moment. Best scenario is to trade it in before it fails, so you get your $3k out of it rather than the $300 from a junk yard to buy it. Personally I would continue to drive it while looking for a better priced part or another vehicle to pull one from that has low miles and clean differential fluid.