I was a full-time student, and my mom paid for my college. Do I still claim it, also?

No.  If your mom claims you as a dependent, she is the only one who claims the tuition credit.

There's a new urban myth among college students that says they can get a $1000 from the government just for filing a tax form. For most of them, they simply aren't eligible. A student, under age 24, is only eligible for the refundable portion of the American Opportunity Credit if he supports himself by working. You cannot be supporting yourself on parental support, 529 plans or student loans & grants. You must have actually paid tuition, not had it paid by scholarships & grants.  It is usually best if the parent claims that credit. 

You cannot claim a credit if you are, or can be, claimed as a dependent by someone else.

Note that it doesn't matter which one of you paid the tuition. The person claiming the student's dependency claims the tuition credit.