Battery, Alternator, or Starter problems

I have never had any issues with my car starting, or any battery issues or anything of that sort. This morning, I left for work and everything started fine. I got to work, was here for about ten minutes and then left again to get gas and breakfast. I pulled into the gas station, turned my car off, pumped my gas, and when I went to get back into my car, it would not start. The lights, radio, etc would all come on, but my engine would not turn and it made a clicking sound. Someone gave me a jump, and my car started up immediately. I let it charge, drove around, got my breakfast, and returned to work. About forty five to fifty minutes later, I left again to go to Auto Zone to have my battery tested. My car started fine, I made it to Auto Zone and went in (after shutting my car off). They came out, tested my battery and it said 100%. They had me turn my car on to test my alternator, and this passed as well. I drove back to work thinking it was just a fluke. Roughly two hours later, I went back outside to see if my car would still work and it was the same thing as this morning: it would not start, made the clicking sound, and lights/radio/etc were all still on, however would shut off when I turned the key. If the key was in the ignition and turned slightly, this all would turn on, but once I turned it to start the car, they would dim/turn off. It once again had the clicking sound as well. I got another jump, and my car started right away. We let it run for a good twenty minutes. I turned it off, waited a few minutes, and turned it right back on with no issues. I am not sure if my battery needs replaced, or if it could be related to my alternator or starter?