The most obvious relationship that is involved in this covenant is the relationship between the two married people. They are making a covenant to stay together in an exclusive relationship for their lifetimes.
A second relationship that is involved in many marriage covenants would be a relationship between each partner and his or her in-laws. When a couple marries, each partner is becoming a part of the other’s family. They are making a covenant to treat one another accordingly.
Since you use the term “covenant,” there is the implication that this is a religious agreement. There is, then, also a relationship between the partners and God. They are promising God that they will remain in an exclusive relationship for their lifetimes.