Graduate students are employees of the university. There are usually other unions at universities for employees, and undergraduate students working in universities can sometimes be in those unions as well, but graduate students as employees face some unique circumstances due to their employment and as a result often form their own union.
Traditionally for better or for worse, undergraduate representation comes by way of a student council, with a single person elected to a chair on the larger university advisory council (the structure can be different depending on the university, I'm generalizing).
There’s a distinction in the US between fully-funded PhD graduate students (who receive a stipend contingent upon research/TA work in an employment relationship) and professional/masters students who pay tuition, even though the latter group is also referred to as graduate students.
Note that for instance if you're there with a fellowship and aren't getting a partial RA/TA assignment as well then you're not eligible. I know that some masters students pay for their tuition, so they might not be eligible either since they're not getting a stipend.
Traditionally for better or for worse, undergraduate representation comes by way of a student council, with a single person elected to a chair on the larger university advisory council (the structure can be different depending on the university, I'm generalizing).