For what rare events was the LHC built? I understand that for low cross-section events a very high luminosity is necessary in order to obtain enough data to produce meaningful statistics. That is why the LHC was built.
But which are these event which we are interested in? What are the events which would hint at new physics or would confirm theoretical models?
 A: The history of high energy physics is in the words "high energy" . There are two ways to get it, building higher and higher energy accelerators  or studying cosmic rays, which last has answers in another question. 
Accelerators are of two types, those creating beams of particles that fall on fixed targets, and colliders, having two beams collide. All studies have yielded the enormous amount of data that one can find in the particle data book, a plethora of resonances and particles  that are encapsulated the Standard Model of particle physics. 
Colliders provide higher center of mass energies and have been for some time  the center of studies for the behavior of higher and higher energies at scattering. The reason is that the standard model fits well low energy data but also predicts data for higher energies, as for example the existence of the Higgs. Also the SM is a gauge that will show if something unusual is happening at higher energies like manifestation of supersymmetric particles etc. by checking crossections and branching ratios against SM calculations .
Colliders come into two physics types  and two geometry types: hadronic, i.e. proton antiproton colliders or proton proton colliders, which is scattering a bag of quarks and gluons (proton) against another bag of antiquarks and gluons (antiproton) or also quarks and gluons (proton), and leptonic, scattering electrons on positrons and watching the fall out.  They can be linear one off beam collisions, or circular colliders the beam increasing in energy by a lot of rotations around the ring.
Leptonic collisions are much more accurate experimentally and theoretically, as the vertices in the Feynman diagrams are simple and computable with fewer assumptions than for hadronic ones.
I have heard that Feynman defended leptonic experiments by saying" if you want to study the interior of a watch you do not throw one watch against another and study the wheels coming out. You take a screw driver ( in this case it was neutrino scattering on a target, I think but it holds for all leptonic collisions).
The one of the drawbacks of leptonic circular  colliders is also their advantage: the known fixed energy at center of mass can explore systematically a phase space with great accuracy and less modeling but the spread of energies is limited.  It is also hard to accelerate electrons and positrons in circular colliders to high enough energies due to synchrotron radiation that degrades the energies of the beams. 
The drawback of the hadron colliders is that a lot of modelling has to enter since it is a scatter of many on many elementary particles but because of the high luminosity achievable they are great for new discoveries. The Z and W were found at a proton antiproton collider at CERN and then were explored in great detail in the leptonic collider LEP, the results of which nailed the parameters of the SM.
Already there are proposals for a higher energy leptonic collider to sit on the Higgs and study with accuracy its branching ratios etc. As well there are plans for an international  linear collider  at high energies after the LHC gives all the hints it can for where to look for new physics.
Hadronic colliders are discovery machines. Leptonic are for nailing down accurately the parameters of the theoretical models.
A: As pfnuessel said in his comment: The first thing to look at was the Higgs - there were hints from LEP and Tevatron, but no evidence, so the LHC was designed that the (SM-)Higgs has to be seen, if it exists.
And for everything beyond the Higgs - we don't know! There are various theories, e.g. the different flavors of super-symmetry and others, but they all have different predictions. If any of these theories is right there must be something - havier SUSY-partners of the standard model particles for example. But as there are many different theories it's not a very targeted search, its more a scan over a broad energy range not reachable in pre-LHC-times - we'll see what the experiemnt-collaborations will stumble upon, and which theory will be best to describe the results...
