A Groebner basis is a specific generating set for an ideal or submodule, often used to compute other information, such as numerical invariants, operations such as intersections and ideal quotients, syzygies, etc. In Macaulay2, Groebner bases are computed behind the scenes when needed, and the Groebner basis is cached in case it is needed later.
See
GroebnerBasis for additional common operations and a comprehensive list of all routines in Macaulay2 which either take Groebner bases as arguments, or return one.
In Macaulay2, Groebner bases may be computed for ideals and submodules over the following rings:
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ZZ -- Hermite normal form
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fields -- Gaussian elimination
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polynomial rings over a field or over the integers (including skew commutative multiplication)
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quotients of such rings
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Weyl algebras
Groebner bases of ideals in polynomial rings over other polynomial rings are also allowed.