Assn. is a written short form of association — a group of people who join together for a job, a sport, a trade, or a cause (a teachers’ association, a medical association). You almost only see it in print, in names and addresses, where space is tight: “the American Heart Assn.” People do not say “assn” out loud; they still say “association.” Though actually, English is messy with these shortcuts — Assoc. is another written short form of the same word, and Ass. can mean assistant, which is a different thing, so read the full name if you can. In Taiwan the Chinese will usually just say 協會; the English abbreviation turns up on letterheads, papers, and US organisation names.