The most well-known missing piece of the Standard Model is the Higgs Boson. The LHC has been built to discover this state which can be produced by gluon-gluon fusion or quark-antiquark annihilation. If the Higgs exists, then it must have siblings (more Higgs bosons) or we must live in a very finely-tuned (unlikely) universe. In either case, the hunt for the Higgs and the determination of its properties will be great fun!
One of the most popular alternatives to the Standard Model is a class of models called Supersymmetry. Supersymmetry requires that every spin 1/2 fermion have a spin 0 boson superpartner and every spin 1 boson have a spin 1/2 fermion superpartner. Supersymmetry also has a rich Higgs boson sector. If these ideas are correct, then the production of heavy gluinos (spin 1/2 partners of the gluon) should be produced copiously at the LHC!
Another promising approach is based upon Extra Dimensions which could result in new TeV scale states.