What is the right expression: vacation days or days off?
Both expressions are grammatically correct. They mean slightly different things, however.
Vacation days generally refers to paid vacation, which are days for which you are paid to work, but don't actually come in. In most organizations, vacation days are usable at the employee's discretion, up to a certain yearly limit.
Days off is a more informal phrase that includes a variety of kinds of paid-not-to-work days, including sick leave, maternal/paternal leave, floating holidays, national holidays, etc. Vacation days are a subset of days off.