2003 Mazda Protege Mom hooked up Jumper cables wrong

The battery went dead on my 87 ford f150 so I tried to jump it off with my 2003 mazda protege. I hooked to cables up to the car first and then started it. My mom was in front of the truck and she started to connect the jumper cables to the truck and a large spark flew from the battery and the mazda died. I jumped from the car to check on her and she was ok and it turned out she had crossed the jumper cables. I got the truck started a little while later and I jumped off the Mazda. It started and everything worked on it so I unhooked the cables from the car and the truck. I sat down in the car, and turned on the headlights and it died, and would not start again. We jumped the car off once more, I turned the headlights on while the car was still connected to the battery cables and they came on. Took the cables off, turned on the AC and the car died once again. I think it is the Alternator but I am not sure. Can someone help me please?