A subgroup of a group is called normal if the sets of left and right cosets of the subgroup coincide, i.e. if $H$ is a subgroup of $G$, then $gH=Hg$ for all $g\in G$.
An equivalent definition is that if $N$ is a normal subgroup of $G$, $(N \vartriangleleft G)$, then:
$$\forall n\in N, \forall g\in G, gng^{-1}\in N$$