These are two verbs which are often confused.
"Approve of" means "think someone / something is good (and therefore give permission for someone to do something)". Here are some examples:
1) My father approves of my new boyfriend.
2) My parents approve of our marriage.
The verb "disapprove" has the opposite meaning to "approve":
3) I disapprove of young people staying out late at night.
4) I disapprove of that kind of language.
The verb "approve" is far less common than "approve of", and is also rather a formal word:
5) Parliament approved the budget.
6) The committee approved the new project.
As sentences (5) and (6) show, the meaning is "officially or formally give permission for something / officially or formally recognize something".
give permission for someone to do something：人が何かをする許可を与える