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. 

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 $\rho(g)$.
