$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\GL}{\mathrm{GL}} \newcommand{\Q}{\mathbf Q} \newcommand{\F}{\mathbf F} \newcommand{\s}{\mathrm{s}} \newcommand{\nr}{\mathrm{nr}} \newcommand{\t}{\mathrm{t}}$

Let me explain Serre's definition in the case of a finite extension $K$ of $\Q_p$ ($p$ prime). In his notation, $K_\s$ is an algebraic closure of $K$, $K_\t$ is the maximal tamely ramified extension of $K$ in $K_\s$, $K_\nr$ is the maximal unramified extension of $K$ in $K_\t$, and $I_\t=\Gal(K_\t|K_\nr)$. The common residue field $k_\s$ of $K_\s$, $K_\t$, and $K_\nr$ is an algebraic closure of the finite field $\F_p$.

So inside $k_\s$ you have all the finite extensions $\F_{p^n}$ of $\F_p$, with norm maps $\F_{p^m}^\times\to\F_{p^n}^\times$ whenever $n|m$. The first thing to observe is that there is a natural identification $\theta$ of $I_\t$ with the inverse limit of this system of norm maps (prop. 2)

As such, for each power $q=p^n$ of $p$, there is a natural projection $\theta_{q-1}:I_\t\to\F_q^\times$, which you might call *the* fundamental character of level $n$. When you compose it with some power $\varphi^i$ of the automorphism $\varphi:x\mapsto x^p$ of $\F_q^\times$, you get all the other fundamental characters of level $n$. There are thus $n$ fundamental characters of level $n$, namely $\varphi^i\circ\theta_{q-1}=\theta_{q-1}^{p^i}$ for $i\in[0,n[$.

Here is how to think of $\theta_{q-1}$ concretely. Take a uniformiser $\pi$ of $K_\nr$ and adjoin its $(q-1)$-th root $\root q-1\of\pi$ to get the extension $L_{q-1}$ of $K_\nr$ in $K_\t$; this extension does not depend on the choice of $\pi$. On the one hand, the group $\Gal(L_{q-1}|K_\nr)$ is a quotient of $I_\t$. On the other hand, $\Gal(L_{q-1}|K_\nr)$ is naturally isomorphic to the group $\mu_{q-1}$ of $(q-1)$-th roots of $1$ (under $\zeta\mapsto(\root q-1\of\pi\mapsto\zeta.\root q-1\of\pi$)), which is naturally isomorphic (under reduction modulo $\pi$) to the group $\F_q^\times$. The character $\theta_{q-1}$ is just the composite
$$
I_\t\to\Gal(L_{q-1}|K_\nr)\to\mu_{q-1} \to\F_q^\times.
$$

I'll leave it to the experts to elucidate the role of fundamental characters in Serre's modularity conjecture (now a theorem). I believe they serve to define the *optimal weight* of a cuspidal eigenform from which a given odd irreducible representation $\Gal(\bar{\Q}|\Q)\to\GL_2(\bar{\F}_p)$ arises.