Informally, a matrix grammar is simply a grammar in which during each rewriting cycle, a particular sequence of rewrite operations must be performed, rather than just a single rewrite operation, i.e. one rule "triggers" a cascade of other rules.
Vector grammars are closely related to matrix grammars, and in fact can be seen as a special class of matrix grammars, in which if m \in M, then so are all of its permutations p(m).