First of all, in prokaryotes, the beta clamp is a dimer. A dimer is made from two [identical] monomers; in this case the monomers are [complexly folded] protein strands.
Secondly, these two dimers form two halves of a Ring - this Ring surrounds or encompasses the Dna strand, thereby forming a clamp that, while securely connected to the Dna strand, can easily move 'laterally' along the DNA strand.
All of the large enzymes that Act upon and Process Dna use the beta clamp as The-Go-Between point of attachment.
In eukaryotes, the biochemical moiety that performs the same function as the beta clamp is a trimer and is called PCNA - the Proliferating Cell Nuclear Antigen.