Mazda Repair: Problems with Car, test batteries, good shape

Question
Today my battery light came one, than I drove about 15 mins and the radio turned off than the whole car started to turn off.  I tried to jump it but once I removed the plugs the car shut off.  After I let the car sit for 4 hours, the car started but the battery light was still on.  I also noticed that the cooling indicator was on the C when I first started the car than went to the halfway mark than the car started to shut down again and the indicator went to C than shut down.  Can you tell me whats going on?

Answer
your battery isn't holding a charge or your alternator is faulty when the car is running perform a load test if you have a canadian tire in your area they normally test batteries for free if the battery comes back ok chances are you have a bad alternator or a part that regulates the alternator have bout tested before replacing the either of them it might not me either of them also check that the connection are tight and clean and the wires are in good shape