My car electrical system is acting up, what is causing it?

I went to the gas station filled up the tank with gas, I got back in car to start it nothing. Lights, radio, wipers, horn worked fine. The tow truck came and though I needed a jump but it did not. The engine would no start or even crank, just a click sound. I drive a 97 plymouth breeze. I had the starter taking out and examined it was fine. Could it be the alternator? Because now the car lights are getting dimmer and dimmer and my car alarm barely works. But when I jump the battery the lights and alarm comes back strong, but dies a couple of days later. what is the problem? I miss my car, it will not start.