I don't get the motivation for calculating Grobner bases. What's good by computing a Grobner basis for an ideal of $k[X_1,...,X_n]$? Moreover, is there any theorem whose proof relies on the use of Grobner basis?
I found "monomial ordering" quite useful while proving the fundamental theorem for symmetric polynomials, but I have not seen yet when Grobner basis plays an important role.