Allow a late comer to add a few complements to the answers given by @Kenny Lau and @Mathmo 123. Actually, the Weil group $W_K$ of a local or global field $K$ is not merely a group, but a triple $(W_K , \phi, {r_L})$ , where $L$ runs through the finite extensions of $F$, which must satisfy a certain number of properties coming from class field theory. More precisely, $W_K$ is a topological group, $\phi: W_K \to G_K$ (= absolute Galois group of $K$ with the profinite topology) is a continuous homomorphism with dense image. This means that $\phi^{-1}(G_L)$ is open for every $L$, and $\phi$ induces a bijection between the homogeneous spaces $W_K/W_L$ and $G_L/G_K$, which is an isomorphism if $L/K$ is Galois. The arithmetic lies in the topological isomorphisms $r_L: A_L \to {W_L}^{ab}$ $:= W_L/{W_L}^c$, where ${W_L}^c$ denotes the closure of the commutator subgroup, and $A_L$ is the multiplicative group $L^*$ if $L$ is local, the idèle class group $C_L$ if $L$ is global. Extra conditions are required, in particular that the composite maps $A_L \to {W_L}^{ab} \to {G_L}^{ab}$ coincide with the reciprocity homomorphisms of CFT, and the natural map $W_K \to pro.lim W_{L/K}$ is a topological isomorphism, where $W_{L/K}:= W_K /{W_L}^c (\neq W_K/W_L)$.
The main work consists in proving the existence and uniqueness (up to suitable isomorphisms) of the Weil group. This is done in detail in Artin-Tate, CFT, chap.14. One key point is thm.5, p.245, which gives canonical isomorphisms $A_K/Norm_{L/K} A_L \cong W_K/W_L{W_K}^c \cong G(L/K)^{ab}$ if $L/K$ is Galois, and justifies a first motivation of the construction : "[This] shows that the entire theory of the reciprocity law is contained in the theory of Weil groups (...) The reciprocity relationship between levels $A_L$ and Galois groups $G(L/K)$ becomes easy to vizualize when one identifies $A_L$ with $W_L/{W_L}^c$ (by means of $r_L$) and $G(L/K)$ with $W_L /W_K$ (by means of $\phi$). In this way, all the facts are wrapped up in one neat non - abelian bundle, namely a suitable Weil group" (p.246). Unfortunately for the layman, chap.14 of Artin-Tate is written in the language of "class formations", which is itself introduced in chap.13 as an axiomatization of the cohomological properties of CFT. So it is perhaps not superfluous to outline the construction of $W_F$ in two particular cases.
1) $F$ is a local $p$-adic field : This case is a relatively simple application of local CFT. The residue field $k$ of $K$ is a finite field of order $q$= a power of $p$ and its absolute Galois group $G_k$ is topologically isomorphic to the profinite completion $\hat {\mathbf Z}$ of $\mathbf Z$ , topologically generated by the Frobenius automorphism $Frob_k:x \to x^q$. It is classically known (see e.g. Cassels-Fröhlich, ANT, chap.1, coroll.2 of thm.2 , p.28) that the maximal unramified extension $K_{nr}$ of $K$ has Galois group isomorphic to $G_k$, hence an inertia exact sequence $1 \to I_K \to G_K \to G_k \cong \hat {\mathbf Z} \to 1$, where $I_K$ is the absolute inertia subgroup. Take $W_K$ = the inverse image in $G_K$ of $<Frob_k>$, so that $W_K/I_K \cong \mathbf Z$, and $\phi$ = inclusion. The topology in $W_K$ will be that for which $I_K$ gets the profinite topology induced by $G_K$, and is open in $G_K$. Consider the local reciprocity map $\theta_K:K^* \to {G_K}^{ab}$. By prop.2 of Cassels-Fröhlich, chap.6, p.141, the composition $K^* \to {G_K}^{ab} \to G_{K_{nr}} \cong \hat {\mathbf Z}$ induces the valuation map $v:K^* \to \mathbf Z$, whose kernel is the group of units $U_K$, hence $\theta_K$ provides a commutative diagram linking the valuation exact sequence to the abelian inertia exact sequence $1 \to J_K \to {G_K}^{ab} \to \hat {\mathbf Z}\to 1$ [I wish I could type the diagram !]. The so called "existence theorem" of local CFT states that $\theta_K$ induces an isomorphism $U_K \cong J_K$ (Cassels-Fröhlich,thm.3a, p.144). From this and the reciprocity isomorphism $A_K \cong {G_K}^{ab}$, the desired property for the maps $r_L$ follows easily.
2) $F$ is a number field : This case is much more involved. At the present time, there exists no direct approach as in 1), the only known construction being somewhat roundabout. Let $L/K$ be Galois with group $G$, and consider the maximal abelian extension $L^{ab}/L$ with Galois group $A\cong C_L/D_L$, where $D_L$ is the connected component of $1$ in $C_K$. By maximality, $L^{ab}/K$ is also Galois, and the extension of groups $1\to A \to G(L^{ab}/K) \to G \to 1$ is described by a 2-cohomology class $\epsilon \in H^2(G, A)$. Besides, define $W_{L/K}$ by the extension of groups $1\to C_L \to W_{L/K} \to G\to 1$ corresponding to the CFT fundamental class $u\in H^2(G, C_L)$, and the Safarevic-Weil theorem asserts that, under the natural map $H^2(G, C_L) \to H^2(G, A)$ induced by reciprocity, the class $u$ maps to the class $\epsilon$ . This gives an obvious commutative diagram linking the two exact sequences of group extensions, hence in particular a surjective map $\phi_L: W_{L/K} \to G(L^{ab}/K)$, with kernel $D_L$ (NB: this also works word for word in the local case, but then $D_L$ is trivial). One finally defines $W_K$ as being $pro.lim W_{L/K}$, and $\phi: W_K \to G_K$ is surjective, with a huge kernel $D_K$ isomorphic to the direct product of $\mathbf R$, $r_2$ circles and $r_1+r_2-1$ solenoids. But, as stressed by Weil (1951), what is actually sought is a Galois-like interpretation of $W_K$ (Weil, 1951).
Thus a second motivation is the study of (continuous) representations of the Weil group $W_K$. In support of the idea that the Weil group behaves like a Galois group, the L-functions attached by Weil (1951) to representations of $W_K$ include as special cases the abelian L-series of Hecke, made with "Grössencharakteren" (=continuous homomorphisms $C_K \to \mathbf C^*$) and the non-abelian L-functions of Artin, made with representations of $G_K$. And beyond lies the Langlands program, but this is outside our scope ./.
