By deep, I mean that our relationships with other followers of Jesus Christ need to move beyond, "Hi. How are you?" "Fine, thanks." When we merely talk about the weather or sports or TV, we are not connecting in the way that God intends for us. Far too many church relationships stay in the shallow end of the pool where the "No Diving" signs are posted. But God wants us to swim out into the deep end and dive together.
Deep relationships develop through self-revealing conversations. They come about when people open up and share about what is really going on in their lives. It is not possible nor advisable to "go there" with everyone, but having a couple of close friendships where you can open up and be honest and vulnerable is an essential element of living as a follower of Jesus Christ.
By meaningful, I mean that our relationships with other followers of Jesus Christ are necessary and vital. There are no disposable people in the church. There is no relationship that we can shrug our shoulders about with an "oh well, I don't need them anyway" kind of attitude. In Matthew 18, right before Jesus told the "rules" for repairing church relationships, he told the story of the lost sheep, reminding us that every member of God's flock is worthy of an all-out search to restore our relationship with them. Meaningful relationships are the ones I cannot live without, the people for whose friendship I crave. And church must be an environment that fosters them.