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1. officially agree to or accept as satisfactory:

〈人･組織などが〉〈計画･考えなど〉を(公に)承認する, …に賛成する

e.g. the budget was approved by Congress

〈公的機関などが〉〈製品など〉を認可する; …をよしとする (!しばしば受け身で)

e.g. (as adjective approved) : an approved profit-sharing plan.

no object believe that someone or something is good or acceptable:

〖approve (of A)〗 〈人が〉(A〈行動･提案など〉に)賛成, 同意する; (A〈人･物〉について)認める, よしとする (!進行形にしない) (↔ disapprove)

e.g. I don't approve of the way she pampers my father and brothers.

2. archaic prove; show:

e.g. he approved himself ripe for military command.

Middle English: from Old French aprover, from Latin approbare (see approbate). The original sense was ‘prove, demonstrate’, later ‘corroborate, confirm’, hence ‘pronounce to be satisfactory’ (late Middle English).