I am looking at these notes, http://www.eecs.berkeley.edu/~luca/books/expanders.pdf
On page 37, Lemma 5.16, the notion of "character" defined seems to be any map from the finite Abelian group to $\mathbb{C}$
On page 38 in section 5.3.1 the "characters" of $\mathbb{Z}_n$ seem to be maps $\chi_r : \mathbb{Z}_n \rightarrow \mathbb{Z}_n$ such that $\chi_r (x) = e^{i\frac{2\pi r x}{n} }$ for each $r \in \{ 0,1,2,3,..,(n-1)\}$
On page 38 in section 5.3.2 the characters of the group $\mathbb{Z}_2^d$ seem to be maps $\chi_r : (\mathbb{Z}_2)^d \rightarrow \mathbb{Z}_2$ as $\chi_r(x) = (-1)^{r.x}$ where $x$ is thought of as string of length $d$ of $0$s and $1$s and for each $r \in \{0,1\}^d$
(..here the set $\{ 0,1\}^d$ is being thought of as a group under bit-wise-xor operation - but what is the inverse?...)
- The last two examples don't seem to be maps into $\mathbb{C}$ as the first definition claimed.
And how were these maps derived? Why these maps and not anything else?
What have these got to do with the notion of "characters" of a representation of a group?
On page 35 he defines characters carefully, and in a standard way. The maps on page 38 give explicit constructions for the characters of the cyclic group, and the characters of the group $\mathbb{Z}_2^n$. In each case you need to check that the given map is a group homomorphism. In the binary case each element is its own inverse under addition. The character values for the cyclic group are complex numbers of norm 1. For the binary group the image is the group $\{-1,1\}$, again a subset of the complex numbers.
As for where these characters come from, the earlier pages give a good idea.
These characters considered here are, for abelian groups, exactly the usual characters in group representation theory.