Vantage point of character theory I am not sure whether I can frame my question properly, or whether at this point my understandings permit me to comprehend the perspectives of the answers to come, but somehow I find it pretty amazing that while doing representation of finite groups over characteristic 0 fields, the character of representation plays such an important role. 
If I look at it separately the trace of a matrix hardly reveals anything about the matrix except if the matrix is diagonal. Does the fact that there exist bases such that I can make $ \rho (g) $ diagonal ( for nice underlying fields at least) for any element g of the group G make a significant difference in our considerations? 
Also, characters being equal implying equivalence of representations is such a stunning fact. Is there any intuitive basis for this? 
And lastly what kind of information comes under the realm of character theory and what is purely representation theory's realm?
I apologize for such a vague question, and reading representation theory for the first time surely I don't understand things too well. But any motivation towards the perspective and vantage points of the subject will be very beneficial. 
 A: Here is how I think about it.
Representation theory takes abstract groups and embeds them into matrix rings, which makes them more concrete. When the group is expressed as matrices, we can use whatever matrix tricks we have learned to help analyze the group.
One tool for matrices is the trace map, which is a good example of an invariant. When you express a linear transformation as a matrix, you have to pick a basis, and that affects which matrix you get. All of the possible matrices for your transformation are conjugate to each other, but in fact their traces are all the same! That's why the trace is referred to as an invariant. 
Even though it seemingly carries very little information, it actually does carry a good bit of information that can be used as a litmus test. With a simple trace computation, you can tell if two matrices are not conjugate immediately.
So, to think about character theory, you can just think of it as representation theory with the trace map applied. Some important information is distilled and carried by the trace map from the representation down to the characters. The fact that distinct characters come from distinct representations means that not too much information is lost in the transition. Things are "boiled down," but not so boiled down that you would confuse two representations.
What is gained by this passage to characters? Presumably characters are easier to handle, or the resulting data is easier to use, at least in their applications. I wouldn't say that the material is divided into the "realm of representation theory" and "the realm of character theory," it's just that some things are more quickly obtained by talking in the character language.
A: I know I'm digging up an old question, but one seems to have brought up this point, although darij grinberg did briefly allude to it in the comments:
For any fixed $g \in G$, knowing the trace of $\rho(g)$ doesn't tell you much. However, the character contains the additional information of the trace of $\rho(g^k)$ for all $k$. If $\lambda_1, \ldots, \lambda_n$ are the eigenvalues of $\rho(g)$, then $$\operatorname{Tr}(\rho(g^k)) = (\lambda_1)^k + \cdots + (\lambda_n)^k.$$ By Newton's identity, knowing this for all $k$ allows one to recover the data of the multiset $\{\lambda_1, \ldots, \lambda_n\}$. Since $\rho(g)$ is diagonalizable, it follows that you know the similarity type of $\rho(g)$.
