I hadn't used my car in a week. Normally it always starts. Then one day after one week it didn't start. I got a jump from my neighbor and i was able to drive around for more than 20 minutes. I stopped for about 10 minutes, turned on the car and was able to drive home. Next morning i can't start my car, I replaced the alternator about 2 years ago. Could it be the battery ?
It could simply be the battery needs a complete charge, or it could be the battery or a parasitic draw. You need to have the battery tested to start. You can see my post here on how to test a battery, or most auto parts stores will do it for you.
20 minutes is not enough to load your battery. If you could start your car after you driven a bit, your alternator is ok. You could try to load you battery, but it could be, that it reached a low point discharge. If so, only new one will help. Also if the battery is more then 5 years old, just change it. Old batteries lose more and more power at the time.