Something that's approved has been officially allowed. The approved uniform for a school, which might include collared shirts and dark pants, is one that is authorized by the principal or headmaster.
Approved things follow the rules — they have been okayed by some authority, whether it's the President of the United States or your parents. Approved subjects for discussion at dinner might be ones your parents have talked about and decided are appropriate. And a movie trailer approved for all audiences is generally rated G by the association that decides what's acceptable for kids to see.