There's an "Other Dependent Credit" of $500 for children over 17 (and other dependents), including full time students, 19-23. This is given to you automatically when you enter a dependent in the personal info section of TT. The money shows on line 19 of form 1040. It is a non refundable credit (if your tax [line 18 of form 1040] is less than $500, you don't get the full $500).
Separately, there is a tuition credit. Parents claim it if the student is their dependent.