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 $G_t=Gal(K_t|K_{nr})$.  The residue field $k_s$ of $K_s$ 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 $F_{p^m}^\times\rightarrow F_{p^n}^\times$ whenever $n|m$.  The first thing to observe is that there is a natural identification $\theta$ of $G_t$ 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 $\theta_{q-1}:I_t\rightarrow F_q^\times$, which you might call *the* fundamental character of level $n$.  When you compose it with some power $\varphi^i$ of the automorphism $x\mapsto x^p$ of $F_q$, you get all the other fundamental characters of level $n$.  There are thus $n$ fundmental characters of level $n$.

I'll leave it to the experts to  write about their role in Serre's modularity conjecture.