This is slightly different in British vs American English.
In the UK we would say "mark" to mean the score on a test. We would only say "grade" to refer specifically to letter-grades scores on a test (A+, A, B, C etc). In the UK we most frequently use "mark/marks". I have heard Americans use both but someone else would need to verify that.
Americans would also say "grade" to mean the school-year class that you are in (1st grade, 2nd grade, 3rd grade etc - each one year older). In the UK we would say "year" for this (Year 1, Year 2, Year 3 etc - each one year older). The school systems in the UK and USA are a bit different, so Grade and Year do not match exactly in number.