Battery or alternator?

Ok I have 1999 Chevy Prism. Now today I went outside to start my car and it wont turn over at all. So I thought I had left something on and my battery died, so I had my neighbor give me a jump, but still it wouldnt turn over. Now Im not sure if I need a new battery or a alternator. I took my multimeter and tested the battery and it read 12.54 volts without the any accessories such as lights, radio, etc on. Then when I turned the lights on the multimeter read 11.26 volts. Im not sure if it a battery, alternator, or maybe even a starter, so does anyone have an idea?

if it was the battery, your lights would dim alot when you turn the key, and you would hear a clicking noise.

if it is your alternater, it wouldnt charge your battery, and your car would run then stall out when the battery was drained.

if it was your starter, it would do what you mentioned. It could also be a loose connection, or a bad ground or something.

but i doubt it is your alternator or battery