A normal subgroup is a special type of subgroup within a group G. A subgroup N of G is considered normal if it is invariant under conjugation by elements of G. This means that for every element g in G and every element n in N, the element gng^{-1} is also in N. Normal subgroups are important because they allow for the construction of quotient groups. When N is a normal subgroup of G, the set of cosets of N in G can be formed, leading to a new group denoted as G/N. This structure is fundamental in group theory and helps in understanding the properties of groups.