I am reading Milne's book, Group theory. And there is a section named 'the linear characters of a commutative group', but I don't see any result that help us understanding group theory itself. Does it help with the classification of finite simple group? Is it involved in any classification theorem of certain type of groups?
A quick google search seems to imply that it is only useful in other branches of mathematics.
By the way, the definition is, a linear character (or just character) of a group $G$ is a homomorphism $G\to \mu(\mathbb{C})$ where $\mu(\mathbb{C}):=\{z\in \mathbb{C}\ |\ |z|=1\}$ with multiplication as group operation.
If it really comes from other branches of mathematics, it is welcome to talk about the origin of character group, just like the name 'solvable group' comes from the Galois groups of finite field extensions that can be obtained by radical extension.
As Qiaochu mentioned, characters of an abelian group generalize to representation theory for possibly nonabelian groups, a deep and beautiful subject.
If you want applications inside group theory, representation theory provides a proof that every group of order $p^a q^b$ is solvable. While proofs avoiding representation theory have been found, it's my understanding that they're much more complicated.
Indeed, there's a proof on the wikipedia page, and you can see how essential it is to understand the characters of $G$. (Moreover, characters in the sense of this wikipedia page specialize to characters in the sense of your question when $G$ is abelian).
These characters (of possibly nonabelian groups again) are also essential for the proof that all groups of odd order are solvable. You can see the wikipedia page on character theory for more.
I hope this helps ^_^