What is the definition of "Augmentation Ideal Filtration"? Let $A$ be an algebra. What is the definition of the Augmentation Ideal Filtration of $A$? Any answer with reference will be greatly appreciated.

In general, a unital $k$-algebra is called "augmented" if it is equipped with an "augmentation", that is a $k$-algebra morphism $\epsilon : A \to k$. Then the augmentation ideal is $A_+ = \operatorname{ker}(\epsilon)$, and $A \cong A_+ \oplus k 1$ where $1$ is the unit of the algebra. The induced filtration is given (as Hanno says in the comments) by $F_i A = (A_+)^i$ (this is the ideal multiplication); then $A = F_0 A \supset F_1 A \supset \dots$ is a filtration of $A$.