to pay attention to someone; to deal with or keep company with someone (listen to them, spend time with them, respond to them).
子どもがずっとまとわりついていて、今日は相手にする余裕がない。
The children have been clinging to me all day, I don't have the energy to deal with them today.
あの人のひどい冗談には相手にしない方がいい。
It's better not to respond to that person's nasty jokes.
上司は忙しくて部下の細かい相談を相手にしてくれない。
The boss is busy and won't deal with the subordinate's minor worries.
to play against; to take on an opponent (in a game, match, or competition).
今週末は強豪チームを相手にする試合がある。
This weekend we have a match against a strong team.
新人だが、彼女は全国大会の優勝者を相手にしても引けを取らなかった。
She's a newcomer, but she didn't lose face even against the national champion.
コンピューター相手にするだけでなく、人と話す練習も必要だ。
You need to practice talking with people as well as with computers.