The character of the group elements
the trace of the matrix representing that group element in the corresponding irreducible representation.
One usage was
With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class-functions, and this yields the orthogonality relation for the rows of the character table.
Thus, the group element's (high dimensional matrix) representation could be mapped to the one dimensional field. But what does this one dimensional field have to do with the group representation's "orthonormality" or not?
Sometimes, the character were used to tell how many "copies" of the representation. But I'm not sure what does that suppose to mean, since the characters could be negative.
What does the character table of the group tell algebraically?