There are many, many answers to this question. In some sense, the LHC isn't so much an experiment as an entire laboratory, and different people will try to use the data it produces in different ways. Here's the most interesting from my point of view:
The Standard Model is based on a symmetry that connects the electromagnetic and weak forces, called (oddly enough) Electroweak symmetry. Now, if you go through the math and figure out what this symmetry implies, you find that if the symmetry is exact then no fundamental particles can have a non-zero mass. We know that the symmetry is not exact, just from the fact that the electromagnetic and weak forces behave very differently at low energies. This leads to an interesting possibility: whatever it is that broke the electroweak symmetry also generated the mass of all fundamental particles. So in some sense understanding the dynamics of electroweak symmetry breaking is equivalent to understanding the origin of mass. That would be very cool. However, there is a very important unanswered question:
What breaks electroweak symmetry?
One of the main motivations for building the LHC (
the main motivation) is to answer this question. The Standard Model proposes that a scalar field, the Higgs, acquires a non-zero vacuum expectation value, and that works mathematically, but we have no direct evidence for such a particle. (We have some indirect evidence, but it's far from a firm conclusion). There are also theoretical reasons to believe that the Standard Model picture is far from complete (search for "The hierarchy problem" or "The gauge hierarchy problem"). For this reason people have proposed many extensions of the Standard Model, including (but not limited to, not by a long shot
): Supersymmetry, Technicolor, Large extra dimensions, Warped extra dimensions, Little Higgs, Fat Higgs, Higgsless, ...
The good news is that there is reason to believe that, whatever the mechanism is, the LHC has a good chance of finding it.
