Augmented Filtrations and Martingales in the Martingale Representation Theorem
The augmentation of the filtration generated by $B$ is typically just called the augmented filtration and is defined using the below construction. It is more commonly referred to simply as the standard Brownian filtration. 


The collection $\mathcal{C}$ of all sets of probability $0$ in the sigma-field $\sigma\{B_{s} \: : \: s \leq t\}$
The collection of null-sets, $\mathcal{N}$, of all $A$ such that $A \subset B$ for $B \in \mathcal{C}$. It is assumed $\mathbb{P}(A)=0$ for all such $A \in \mathcal{N}$.


Then the augmented filtration for Brownian motion to be the filtration given by $\{\mathcal{F}_{t}\}$, where for each $t$ we take $\mathcal{F}_{t}$ to be the smallest sigma-field containing $\sigma\{B_{s} \: : \: s \leq t\}$ and $\mathcal{N}$.

The point of this is to endow the filtration with some properties which are considered helpful.

The relationship to martingales is simply that $X_{t}:=\mathbb{E}(X\:|\:\mathcal{G}_{t})$ is a martingale.