Partner, Companion, Opponent
noun
Remember how 手 can mean a "doer"? Now, what do you think your mutual doer might be? It’s someone who shares the same activities or tasks with you. This person could be your partner, companion, or even an opponent in certain situations.
あいて
The readings are the kun'yomi readings. For 手, that's no problem for you. For 相, that's more of an issue since you learned the on'yomi reading for it. Think of it this way: Your partner wouldn't be your partner without you. Your partner is: I (あい) and you. Remember that as you gaze into your partner's eyes. "I and you."
今回のゲームの相手は強そうだ。
The opponent of this game seems to be strong.
母の話し相手は、ペットの毛虫です。
My mother's companion for chatting is her pet caterpillar.
フラッシュモブでプロポーズされる相手のことも、きちんと考えなよ。
How about thinking carefully about the person you're proposing to with a flash mob?