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.
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."
相手にする
to listen to someone, to make someone one's opponent
相手になる
to become a partner
相手に言う
to say to one's partner
相手の心
partner's heart
相手の人
partner
相手の目
partner's eye
相手を見つける
to find a partner
相手をえらぶ
to choose a partner
相手をする
to be a partner for someone
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。
Take the time and properly consider the person you're proposing to in the flash mob too.