A graduate school (or grad school) is a school that gives advanced academic degrees, such as master's degrees (M.S. or MBA ...) and doctoral degrees (Ph.D. or L.L.D. or D.D.S. etc.). A graduate student (or grad student) is a student who goes to the school. Also, in British English, the phrase "postgraduate student" is more common.
The term "graduate school" could be confusing because the verb "graduate" is also used to mean completing high school education or a bachelor's degree (known as an "undergraduate degree"). The verb infinitive "to graduate" is used for all those types of schools, but the meaning of the adjective "graduate" is changed to mean "postgraduate" in the term "graduate school". The small change in meaning can be viewed with an air of sophistication, where experienced students are often more aware of the issue, while novice students might not be aware, as being more naive.