If you're pursuing a degree beyond a bachelor's, you're considered a graduate student. This can include working to earn a master's degree or doctorate/Ph.D., or enrollment in a professional program such as law school and medical school.
It gets tricky if you tackle a joint undergraduate/graduate program since some schools deem you a graduate student once you start taking graduate-level classes.
Also, if you're getting a second bachelor's degree, you could be considered a graduate student when it comes to receiving financial aid. This is determined on an individual basis, so it's best to talk to your school about these unique scenarios.