God's very being is love. By sending his only Son and the Spirit of Love in the fullness of time, God has revealed his innermost secret: God himself is an eternal exchange of love, Father, Son and Holy Spirit, and he has destined us to share in that exchange. (Catechism of the Catholic Church, [CCC], par. 221)
The mystery of the Trinity is the central tenant of our faith, but trying to wrap your mind around how God can be one-in-three and three-in-one is a good way to get a headache.
We are made in the image and likeness of God, and we can get a small glimpse of the communion and unity that exists in the Godhead by observing human life. The sacrament of marriage has an awesome power: two people become one. In total love and self-giving to the the other, their love actually takes on flesh and becomes another person in the form of a child. In a small way, family life mirrors the Trinity where the love and self-giving between the Father and the Son is so real as to become another Person, the Holy Spirit.
Religious are also called to incarnate the union in community that is the Holy Trinity. Religious living together in community are bonded together in love and unity, a true family gathered together in the Lord's name. Made up of individuals, the community has one heart and a common goal, to fulfill God's will as laid out for them in a particular charism.
In creating man and woman in his own image and likeness, God created them for communion. God the Creator, who revealed himself as Love, as Trinity, as communion, called them to enter into intimate relationship with himself and into interpersonal communion, in the universal fraternity of all men and women.
This is our highest vocation: to enter into communion with God and with our brothers and sisters.
(Fraternal Life in Community, 9)