New physics is what we call physics beyond the Standard Model. Those of us who work with colliders, like the Tevatron at Fermilab, are concerned with finding such new physics for a number of reasons. Although one piece of the Standard Model - the Higgs boson accounting for electroweak symmetry breaking, or why things have mass - is not yet found, we have many reasons to expect to find it or something like it in the near future. However, the Standard Model has its problems. The Higgs introduces a number of fine-tuning problems. We don't like to think that a theory must have certain parameters fixed exactly to tens of decimal places in order for things to work. It doesn't seem very natural. (One finds after a while that naturalness is a big reason for doing many things in physics and mathematics.) So we look at ways to explain electroweak symmetry breaking (hereafter I'll call this EWSB for simplicity). One way is supersymmetry, a new type of symmetry relating fermions (particles with half-integer spin) to bosons (particles with integer spin). It stabilizes the divergences associated with the Higgs, eliminating a need for fine-tuning. On the other hand, it introduces around 100 new parameters to be explained. It also predicts a number of particles we haven't observed yet, but which we actively look for. For example, particles like the electron and the quarks now have partners called selectrons and squarks. Bosons like the W and photon have partners called the Wino and photino. Supersymmetry has the advantage that locally-broken supersymmetry necessarily gives a theory incorporating gravity, called, naturally enough, supergravity. Unfortunately, supergravity is perturbatively sick, meaning nasty infinities crop up in the theory which we simply don't know how to get rid of. This led people to superstring theory and M-theory, but I'm over my head talking about those and they're not directly relevant to experimental tests.
So, what other sorts of new physics might we expect to see within the next decade? The Large Hadron Collider should go online in about 5 years, and exciting new things could be seen then if the Tevatron doesn't see them first.
Besides supersymmetry, here are some other things we look for:
Large Extra Dimensions
Proposed fairly recently by Arkani-Hamed, Dimopoulos, and Dvali, this model says that the universe may have extra dimensions as large as tens of micrometers (originally the claim was about a millimeter, but tests have pushed the number continuously lower since). Gravity would extend through all the dimensions, but the Standard Model fields would be localized in a four-dimensional subspace (where we live). This explains why gravity is so much weaker (the so-called hierarchy problem): it is spread out over more space. (The argument uses Gauss's Law, taught in fairly elementary physics classes.) However, it doesn't explain why these spatial scales are so different, so in a way it introduces its own new hierarchy problem.
If these models are correct there could be black holes produced at the Large Hadron Collider! This would be quite stunning as we could actually experimentally probe quantum gravity, but it is fairly unlikely. Don't worry about any danger; if black holes are made in such a situation then there must be many of them made in our atmosphere all the time by cosmic rays. They would then evaporate by Hawking radiation.
There are also the Randall-Sundrum models, which have a "warp factor" in the metric of the extra dimensions that contains an exponential factor. We are stuck on a 4-dimensional brane, and the exponential accounts for the weakness of gravity. This model is more natural and so often preferred, but it could have its own problems (like the existence of a light particle, the radion, which is somewhat hard to explain).
Grand Unified Theories
These theories embed the Standard Model gauge group within a larger group, like SU(5), SO(10), or E(6). From these, we expect new, heavier vector gauge bosons, and other effects. The E(6) models in particular have characteristics resembling supersymmetry in terms of what would be seen in particle accelerators. This is good, as it keeps us from jumping to conclusions too quickly - we need lots of viable alternative theories so that we know both what to look for and what we're looking at when we find something.
"Deconstructing Dimensions"
Those wacky guys behind the extra dimensions model described above started looking at the patterns of Kaluza-Klein modes, or extra particles seen due to the structure of excitations in the extra dimensions. They realized that they could get the same sort of thing from ordinary four-dimensional theories, using tools variously called moose diagrams or quiver diagrams to help construct the theories. No, I'm not making this up. One paper had the title "Deconstructing Noncommutativity with a Giant Fuzzy Moose." But I digress. The interesting thing is, that even though as far as I can tell the original motivation involving the hierarchy problem is lost in such models, they led to new theories with radical new forms of EWSB. And new ways of tackling unsolved problems are always good. Which leads me to my last theory to bring up:
Technicolor
This one's an old idea that's almost dead, but not forgotten. It proposed a new strong force to break electroweak symmetry; instead of the Higgs, we would get all sorts of particles resembling those of quantum chromodynamics, like "technipions." Although the name is fun, it generally proved difficult to make a realistic theory (consistent with experiment) from the idea. But the suggestion that the Higgs is not just a simple scalar particle is still floating around, and it may turn out that this sort of theory, with some changes, is needed in the end.