Matchmaking
noun, する verb
A mediator serves families and even countries by matchmaking for them. They know everyone and everything so to stop family fights and wars, they put two people from each side together to get married and it fixes everything.
ばいしゃく
This is a jukugo word that uses the on'yomi readings of the kanji. You should be able to read this on your own.
私どもの結婚式の媒酌の労をおとりいただき、本当にありがとうございました。
Thank you so much for all your hard work in our wedding ceremony matchmaking.