Despite the impressive successes of the SM, our understanding of the fundamental interactions is far from being complete. Many questions remain unanswered, most notably the origin of the masses of elementary particles. In the SM, masses are a consequence of the dynamics of a new particle introduced ad hoc in the theory, the Higgs boson. This spin-0 particle is supposedly responsible for the masses of all elementary particles. This sole mechanism should explain masses in a range of eleven orders of magnitude, from the neutrino mass up to the top quark mass. This may indicate the need of a dynamical mechanism to explain this large fermion mass hierarchy.
There is also the possibility that the Higgs sector of the SM is in fact an effective description of a more complex dynamics, which may involve new degrees of freedom and possibly new symmetries. One hint in this direction is the so-called gauge hierarchy problem: why is the weak scale so much smaller than the Planck scale of 1016 TeV. Theories involving elementary scalar particles, like the Higgs boson, are not stable under quantum corrections unless there is a symmetry protecting their stability (e.g. supersymmetry). Thus, either such a symmetry (and the new associated particles) is present at the TeV scale or the Higgs sector must not be elementary. Either way, the solution of the hierarchy problem points towards the existence of new physics in the TeV scale and the LHC is the ideal place to investigate this possibility.
There are other important questions arising from astronomical and cosmological observations. For a long time there has been data suggesting the existence of dark matter, whose nature remains unknown. Certainly, a new weakly interacting particle, with mass in the range of tens to hundreds of GeV, can answer this puzzle. The experiments at the LHC might shed some light on any of these problems where the interplay between particle physics and cosmology exists.