Definition of the Filtration of a Module

Question

Let $R$ be a ring, and $M$ an $R$-module. What is the definition of a filtration of $M$? Is it simply a descending sequence of submodules $M_{1} \supset M_{2} \supset M_{3} \supset \dots \supset \dots$. Does it have to be countably infinite or can it be finite? For example, what is an example of a filtration of $M = \mathbb{C}[x,y] / \langle x^{2}y \rangle$ as a $\mathbb{C}[x,y]$ module.