When you do something in cooperation with another person, the two of you do it mutually. You and your sister might, for example, mutually decide to skip the family reunion.
You can use the adverb mutually to describe an action or a connection that you share with someone. If you have a mutually competitive relationship with your best friend, it means that both of you feel a sense of rivalry. And when a decision is mutually beneficial to everyone in town, it helps every single person equally. The root word is the Latin mutuus, which means "reciprocal, or done in exchange."