1Something is vague or unclear - its meaning, attitude, or expression is not clearly defined.
彼の返事は曖昧で、賛成なのか反対なのか分からない。
His reply was vague; I can't tell if he agrees or disagrees.
契約書に曖昧な表現があると、後でトラブルになりやすい。
Vague wording in a contract tends to cause trouble later.
質問には曖昧にせず、はっきり答えてください。
Please answer the question clearly instead of being vague.
記憶が曖昧で、あの日何があったのかよく思い出せない。
My memory is hazy, and I can't quite recall what happened that day.
2Shady or disreputable - having something suspicious or improper about one's conduct; can also describe a disreputable establishment (cf. 曖昧宿).
あの店は曖昧な商売をしているという噂だ。
There are rumors that shop is running a shady business.
彼の曖昧な素性が周囲の不信感を招いた。
His shady background aroused suspicion among those around him.
3"Fuzzy" in mathematics/computer science - treating truth values as a continuous range rather than strict true/false.
あいまい検索を使えば、多少入力を間違えても正しい結果が出やすい。
Using fuzzy search makes it easier to get correct results even with minor input errors.