Why is Wilson's work so relevant in particle physics? I thought that critical phenomena were described by CFTs If I have understood the subject correctly, Wilson correctly explained the second order phase transitions and was able to compute the critical exponents of several of them using the Renormalization Group. 
However, since critical phenomena are described by Conformal Field Theories and the Standard Model of Particle Physics is not I wonder why is Wilson's work so important in our current understanding of the Standard Model.
 A: Some history: In the pre-Wilsonian interpretation of quantum field theory, renormalisability of theories was considered an essential requirement. That is, you should be able to remove all ultraviolet divergences of the theory (that occur in perturbation theory) by absorbing them in a finite number of parameters in the lagrangian. If you could not do that, then the theory was called "non-renormalisable" and considered pathological. The renormalisability criteria placed strong constraints of the theory, determining which finite number of interactions were allowed. The standard model is renormalisable.
One effect of the renormalization process was that coupling constants became scale (energy) dependent, and such running coupling constants were understood to be physical. eg asymptotic freedom is the result of the QCD coupling constant becoming weak at large energies.
Wilson's goal and approach in condensed matter was different. He was interested in studying the behaviour of complicated theories near the second order phase transition where the correlation length becomes very large, that is when the theory is essentially scale invariant. That means that you could study the system via an effective theory near the phase transition point without worrying about the underlying microscopic degrees of freedom. In this approach you write down a simple field theory with the desired symmetries and you use a cut-off $\Lambda$ in all calculations since your theory is only an approximation at low energies $E \ll \Lambda$ (long distance scales). 
So in the Wilsonian approach there are no ultraviolet divergences. But it also meant that you had to include many more interactions in your lagrangian. The coupling constants also became $\Lambda$ dependent and you had a running of the couplings here too. In practice, in the limit $E/\Lambda \to 0$, only a finite number of couplings are important: the ones that are "marginal" and "relevant". The ones that are "irrelevant" are the ones that a particle physicist would have labelled as "non-renormalisable". 
So it was initially a different philosophy and approach in high-energy physics and condensed matter, but gradually high-energy physicists understood that the Wilsonian perspective of effective field theories is applicable to their studies of quantum field theory. 
In the modern perspective, the standard model is believed to be a low-energy approximation of some as yet unknown theory. Although the current model is renormalisable, it is believed that "non-renormaliable = irrelevant" interactions would become important at higher energies, and they would represent new processes. So you can write down what the next possible interaction could be (eg to give neutrinos small masses) and make some predictions. 
Such an effective field theory approach is also useful when you want to make low energy predictions from your high energy theories. 
So, in summary, the Wilsonian perspective of quantum field theories and the renormalisation group is important because it gives them a very physical and intuitive meaning.
Now, some comments about "conformal field theories". A bigger symmetry than just scale invariance is "conformal invariance". In two dimensional space conformal invariance results in an infinite dimensional symmetry group which places strong constraints on a theory and allows many exact results to be obtained in condensed matter systems. It is also important in string theory as the world sheet is two dimensional. Conformal symmetry is less powerful in higher dimensions as the symmetry group is finite and the symmetry is typically broken by quantum effects through the generation of mass scales.  
A: I'm not sure if this will directly answer your question, but perhaps it will be helpful. I will simply quote a few sections from the beginning of Conformal Field Theory by Phillipe Francesco, picking only those which relate to high energy particle physics (CFT's are very important in understanding quantum critical points in condensed matter systems).

Scattering experiments failed to detect a
  characteristic length scale when probing the proton deeply with
  inelastically scattered electrons. This supported the idea that the
  proton is a composite object made of point-like constituents, the
  quarks. . . 
In other words, in this deep-inelastic range, the internal dynamics of
  the proton does not provide its own length scale $\ell$ that could justify
  a separate dependence of the structure functions on the dimensionless
  variables $\ell^2\nu$ and $\ell^2 q^2$. In the context of quantum chromodynamics (QCD,
  the modem theory of strong interactions), this reflects the asymptotic
  freedom of the theory, namely, the quasi-free character of the quarks
  when probed at very small length scales. 
Of course, the quark-gluon
  system underlying the scaling phenomena of deep inelastic scattering
  is thoroughly quantum-mechanical, just like systems undergoing
  quantum-critical phenomena. However, scale invariance manifests itself
  at short distances in QCD, whereas it emerges at long distances in
  quantum systems like the Heisenberg spin chain.

In regards to String Theory:

The first-quantized formulation
  of string theory involves fields (representing the physical
  shape of the string) that reside on the world-sheet. From the point of
  view of field theory, this constitutes a two-dimensional system,
  endowed with reparametrization invariance on the world-sheet,
  meaning that the precise coordinate system used on the world-sheet has
  no physical consequence. . . This reparametrization
  invariance is tantamount to conformal invariance. Conformal invariance
  of the world-sheet theory is essential for preventing the appearance
  of ghosts (states leading to negative probabilities in quantum
  mechanics). The various string models that have been elaborated
  basically differ in the specific content of this conformally invariant
  two-dimensional field theory (including boundary conditions). A
  classification of conformally invariant theories in two dimensions
  gives a perspective on the variety of consistent first-quantized
  string theories that can be constructed . . . 
string scattering amplitudes were expressed in terms of correlation
  functions of a conformal field theory defined on the plane (tree
  amplitudes), on the torus (one-loop amplitudes), or on some
  higher-genus Riemann surface.

And further regarding the operator product expansion (OPE) and conformal bootstrap:

The modem study of conformal invariance in two dimensions was
  initiated by Belavin, Polyakov, and Zamolodchikov, in their
  fundamental 1984 paper. These authors combined the representation
  theory of the Virasoro algebra . . with the idea of an algebra of local operators and
  showed how to construct completely solvable conformal theories:
  the so-called minimal models. An intense activity at the border of
  mathematical physics and statistical mechanics followed this initial
  envoi and the minimal models were identified with various
  two-dimensional statistical systems at their critical point. More
  solvable models were found by including additional symmetries or
  extensions of conformal symmetry in the construction of conformal
  theories.
A striking feature of the work of Belavin, Polyakov, and
  Zamolodchikov . . . regarding conformal theories is the minor role
  played (if at all) by the Lagrangian or Hamiltonian formalism.
  Rather, the dynamical principle invoked in these studies is the
  associativity of the operator algebra, also known as the bootstrap
  hypothesis. . . The key ingredient
  of this approach is the assumption that the product of local
  quantum operators can always be expressed as a linear combination of
  well-defined local operators . . . This is the operator product expansion,
  initially put forward by Wilson . . . The dynamical principle of the
  bootstrap approach is the associativity of this algebra. In practice,
  a successful application of the bootstrap approach is hopeless, unless
  the number of local fields is finite. This is precisely the case in
  minimal conformal field theories. . . 
Following the pioneering work of Belavin, Polyakov, and Zamolodchikov, 
  conformal field theory has rapidly developed along many directions.
  The work of Zamolodchikov has strongly influenced many of these
  developments: conformal field theories with Lie algebra symmetry (with
  Knizhnik), theories with higher- spin fields—the W-algebras—or with
  fractional statistics—parafermions (with Fateev), vicinity of the
  critical point, etc. These developments, and their offspring, still
  constitute active fields of research today and make conformal field
  theory one of the most active areas of research in mathematical
  physics.

