A teaching assistantship involves helping a Professor with Labs and grading. It may involve actual teaching. TA gets a salary which may include tuition waiver and health insurance benefits.
A research assistantship involves helping a Professor with research work. An RA is expected to complete research related tasks as assigned. RA gets a salary which may include tuition waiver and health insurance benefits. If you work as an RA with your own advisor, the RA work could be combined with your own MS and PhD thesis, so it saves some precious time.
Students may be employed by other departments or offices in the University on similar salary and tuition waiver as TA or RA, but they do not teach or conduct research. These positions are called GA or GANT (Graduate Assistant Non Teaching). It is like a part time job. The schedules could be worked out every semester to avoid conflict with studies and exams.