An rdf store holds place for storing the RDF datamodel.
The rdf datamodel is structured as a sequence of: s (subject), p (predicate), o (object)
In principle a table with 3 fields would be enough.
Indexing is done according to the necessities in performance for reading/storage.
A triple store is normaly a synonym for an RDF store.
A quad store holds a fourth attribute above the -s -p -o triple: the graph name.
The graph name corresponds normally with the namespace of the ontology.
Some vendors choose another name for the fourth field but the purpose remains the same.
Related service: ontology development