The most obvious fact about monomials is that any polynomial is a linear combination of them, so they can serve as basis vectors in a vector space of polynomials - a fact of constant implicit use in mathematics. An interesting fact from functional analysis is that the full set of monomials tn is not required to span a linear subspace of C[0,1] that is dense for the uniform norm (sharpening the Stone-Weierstrass theorem). It is enough that the sum of the reciprocals n-1 diverge (the Müntz-Szasz theorem).
Notation for monomials is constantly required in fields like partial differential equations. Multi-index notation is helpful: if we write α = (a,b,c) we can define Xα = X1aX2bX3c and save a great deal of space.
In algebraic geometry the varieties defined by monomial equations Xα = 0 for some set of α have special properties of homogeneity. This can be phrased in the language of algebraic groups, in terms of the existence of a group action of an algebraic torus (equivalently by a multiplicative group of diagonal matrices). This area is studied under the name of torus embeddings.