A presentation of M is a map p so that coker p is isomorphic to M. The presentation obtained is expressed in terms of the given generators, i.e., the modules cover M and target p are identical. The isomorphism can be obtained as map(M,coker p,1).
Since a module M may be described as a submodule or a subquotient module of a free module, some computation may be required to produce a presentation. See also prune which does a bit more work to try to eliminate redundant generators.
For a quotient ring R, the result is a matrix over the ultimate ambient polynomial ring, whose image is the ideal defining R. The entries of the matrix form a Groebner basis.
See also cover.