This is a very important theme in the play, perhaps the most important. Loyalty to one's family and community is considered almost sacrosanct among the Italian Americans who live in this part of New York. It's the glue that holds people together and protects them from outsiders, making the numerous challenges that the community faces marginally less difficult to handle. Loyalty is a source of strength and solidarity for the people who live here, and woe betide anyone who doesn't understand this.
That brings us on to Eddie Carbone. Eddie ultimately comes to grief by betraying the long-established principle of loyalty to one's family and community. His fateful decision to rat out Marco and Rodolpho to the immigration authorities makes him an instant social pariah. In this established Italian-American enclave of Brooklyn, such acts of disloyalty are considered beyond the pale.
In trying to get Marco and Rodolpho deported, Eddie is putting his own selfish needs above those of the wider community. He wants them out of the way, not because he's a fine, upstanding citizen, or because he has a sudden desire to see the immigration laws properly enforced, but because he wants Catherine all to himself. Eddie's loyalty is to himself and no one else.