What are Dirichlet characters?

What are Dirichlet characters? I don't really understand the definitions given by wikipedia or wolframalpha, are they defined to be partially multiplitictive just because that gives them, an euler product representation? Or is there some other reason?

Your question seems to be about both the definition and the motivation behind it. Here is a brief discussion of both.

The definition:

A Dirichet character modulo $N$ is a multiplicative homomorphism $\chi$ from $(\mathbb Z/N\mathbb Z)^{\times}$ (the group of units in the ring of integers modulo $N$) to $\mathbb C^{\times}$ (the group of non-zero complex numbers). It can also be regarded as a function on the set of integers that are coprime to $N$ by defining $\chi(a) := \chi(\text{residue class of } a \bmod N),$ and then its defining properties are that $\chi(a b) = \chi(a)\chi(b)$ and that $\chi$ is of period $N$ (so that $\chi(a)$ really does only depend on the class of $a$ mod $N$).

Sometimes $\chi$ is extended to a function $\mathbb Z/N\mathbb Z \to \mathbb C$ (or $\mathbb Z \to \mathbb C$) by defining $\chi(a) = 0$ if $a$ is not a unit mod $N$ (i.e. if $a$ is not coprime to $N$). This is just done mainly as a technical convenience, so that one can write down formulas (like the formula for the Dirichlet $L$-series of $\chi$) in such a way as to not have to specifically restrict to integers that are coprime to $N$. Note that with this extension of the domain of $\chi$, one still has that $\chi(ab) = \chi(a)\chi(b)$ and that $\chi$ is periodic of period $N$. (Actually, to be precise, one should say that the period divides $N$; if it exactly equals $N$, we say that $\chi$ is primitive.)

The motivation:

As you note, because $\chi$ is multiplicative, the $L$-series of $\chi$ admits an Euler product. This is certainly the fundamental reason for singling out the Dirichlet characters.

But another key point is that, by Fourier theory for finite abelian groups, any function on $(\mathbb Z/N\mathbb Z)^{\times}$ (for example, the function $\delta_a$ which is one on the coset of some fixed integer $a$ coprime to $N$ and zero on all other cosets) can be written as a linear combination of characters. So one can use Dirichlet characters to study any function on $(\mathbb Z/N\mathbb Z)^{\times}$. This is how they get used in the proof of Dirichlet's Theorem on primes in arithmetic progression (where the function $\delta_a$ is the one we need to understand).

Combining these two points, one sees that one can use the $L$-series of Dirichlet characters to analyze the behaviour of mod $N$ functions (such as $\delta_a$) on primes, and this is what is done in the proof of Dirichlet's Theorem.