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to speak or think favorably of; pronounce or consider agreeable or good; judge favorably:
to approve the policies of the administration.
to consent or agree to:
Father approved our plan to visit Chicago.
to confirm or sanction formally; ratify:
The Senate promptly approved the bill.
Obsolete.

to demonstrate; show.
to make good; attest.
to prove by trial.
to convict.

to speak or consider favorably (sometimes followed by of):
Mother didn’t approve of him. The boss wouldn’t approve of the plan. He said that he approved.
Contemporary Examples

The House could defy my expectations and approve a resolution (the Senate almost surely will).
Why Obama’s I-Might-Bomb-Syria-Anyway Stance Could Backfire Michael Tomasky September 2, 2013

They come through on our end in a queue, and we’ll approve one-by-one so Ogi/Sai/Jess have the time to answer them.
Newsweek Live Chat: What the Internet Can Teach Us About Sex The Daily Beast April 24, 2011

And I approve this message because NO ONE from New York or Washington tells ME what to do.
Who’s Mark Pryor Kidding? Michael Tomasky May 30, 2013

However, the Republicans have agreed to approve anyone–anyone!
Reid Crushes McConnell Michael Tomasky July 15, 2013

Unaccountable secret money will do and say things that candidates who must “approve this message” won’t do.
Good Candidates, Bad Election David Frum August 7, 2012

Historical Examples

The children, already sitting in a decorous ring on their low chairs, seemed after the first surprise to approve of Phyllis.
The Rose Garden Husband Margaret Widdemer

Folks pretend to approve of ’em and all the while they’re laughing at ’em up their sleeves.
Dust Mr. and Mrs. Haldeman-Julius

Someone would appear there at the turn of a path and would stand before me and smile and approve.
The Turn of the Screw Henry James

I have written an answer to his letter; will you look at it, and tell me if you approve of it?
Tales And Novels, Volume 3 (of 10) Maria Edgeworth

He was still in that state of mind where the lover hopes that his beloved will approve of his friends.
The Vehement Flame Margaret Wade Campbell Deland

verb
when intr, often foll by of. to consider fair, good, or right; commend (a person or thing)
(transitive) to authorize or sanction
(transitive) (obsolete) to demonstrate or prove by trial
verb
(transitive) (law) to improve or increase the value of (waste or common land), as by enclosure
v.

c.1300, “to demonstrate, prove;” mid-14c., “to attest (something) with authority,” from Old French aprover (Modern French approuver) “approve, agree to,” from Latin approbare “to assent to as good, regard as good,” from ad- “to” (see ad-) + probare “to try, test something (to find if it is good),” from probus “honest, genuine” (see prove).

The meaning extended late 14c. to “to sanction, endorse, confirm formally” then to “assent to (something) as good” (early 15c.), especially in reference to the actions of authorities, parliaments, etc. Related: Approved; approving.