Every polynomial ring in Macaulay 2 comes equipped with an ordering on the monomials. See below for the definitions of all implemented orderings. The default ordering is GRevLex, the graded reverse lexicographic order.
Polynomials are displayed by ordering the monomials in decreasing order. The choice of monomial order can make a difference in the time and space required for various computations, especially Groebner basis computations.