Everybody has heard the sensationalist claims about the alleged dangers of black holes at the LHC, but the real physics rationale behind the LHC and its experiments has been featured much less prominently in the media. So what do physicists actually hope to find with the help of the LHC?
The best answer is probably "the unknown". Since we have never collided particles at energies comparable to that of the LHC, we have no idea what particles can be created in such collisions. (Of course there have been lots of cosmic ray events at these and even at much higher energies, but you can't build a detector around those to study what comes out of the collision.) There might be all kinds of surprises waiting for us there. But of course we theorists have not been lazy while the engineers and experimentalists constructed the LHC, and so there lots and lots of theories, models and ideas around about what the LHC experiments could conceivably observe.
The next best answer is certainly "the Higgs boson". The Standard Model of elementary particle physics describes all of known physics with great accuracy, but one of its crucial predictions, the existence of the Higgs boson, remains unconfirmed by experiment. If the LHC finds the Higgs boson, this will be the final confirmation of the Standard Model, meaning that we would then be sure that we do in fact understand all of particle physics up to energies of many TeV. If the LHC were not to find the Higgs boson, or at least something very much like it, this would mean that all our current ideas about what gives particles their masses and what causes the breaking of electroweak symmetry are completely wrong.
Many physicists also hope that the LHC will find evidence of supersymmetry. Supersymmetry is a kind of symmetry that links bosons and fermions by giving a "superpartner" to every kind of particle, and which would provide an elegant solution to a number of problems that worry theoretical particle physicists, such as why the energy scale of electroweak symmetry breaking is so much smaller than the Planck energy, or why the cosmological constant is so tiny. So far there is no experimental evidence whatsoever that supersymmetry exists, but if it exists, the LHC has a good chance to discover at least some of the superpartners of known particles, which would mean that so far we know only half of the stuff of the universe and thus would open up a huge new field to experimental study.
Whether supersymmetry exists in nature or not, we know that there is a lot of stuff that we don't see, but that exerts a gravitational pull on cosmic scales, and which we call "dark matter". The LHC should probably be able to create the particles of which this dark matter consists, allowing us to study their properties and thus advance our understanding of cosmology.
Physicists working on theories containing additional dimensions beyond the usual four (three of space, one of time) also hope that the LHC might uncover evidence of the existence of these additional dimensions, but that is already far in the realm of the speculative, and getting close to black hole territory on the scale of improbability (though I hasten to add once again, that the very theories that might enable black holes to be formed at the LHC also predict that those black holes will decays almost instantly, and certainly much faster than they could possibly accrete matter).
Ultimately, the most wonderful thing the LHC could discover would be something that no theorist has yet predicted, but no matter what it finds, it will certainly falsify 90% or more (maybe even 100%) of the currently existing theories and models, and that is the real excitement.