What is $p_T$? (transverse momentum?)
The component of momentum transverse (i.e. perpendicular) to the beam line.
It's importance arises because momentum along the beamline may just be left over from the beam particles, while the transverse momentum is always associated with whatever physics happened at the vertex.
That is, when two protons collide, they each come with three valence quarks and a indeterminate number of sea quarks and gluons. All of those that don't interact keep speeding down the pipe (modulo Fermi motion and final state interaction).
But the partons that react do so on average{*} at rest in the lab frame, and so will on average spray the resulting junk evenly in every direction. By looking at the transverse momentum you get a fairly clean sample of "stuff resulting from interacting partons" and not "stuff resulting from non-interacting partons".
There are also advantages related to the engineering of the detector.
{*} Only on average. Indivudual events may involve high Bjorken $x$ particles and be a long way from at rest in the lab frame.
The collisions of protons are complicated, because the proton has a big mess inside. In order to see simple collisions, you want to find those cases where a single quark or gluon, a single parton scattered off another parton in a nearly direct collision. Such collisions are relatively rare, most proton proton collisions are diffractive collective motions of the whole proton, but every once in a while, you see a hard collision.
The characteristic of a hard collision is that you get particles whose momentum is very far off the beam line direction. This is a "high P_T" event. A high P_T electron usually means that an electrically charged parton (a quark) collided with some other parton, and emitted a hard photon or a Z which then produced an electron and a positron. Alternatively, it could mean that a W boson was emitted by the quark, and this produced an electron and a neutrino. Alternatively, it could be a higher order process in the strong interaction, where two gluons produced a quark-antiquark, and one of the quark lines then emitted an electroweak boson, which decayed leptonically.
The point is that any way it happened, the event indicates that a clean hard collision happened between two partons, and this is a useful indication that the event was an interesting one, which will give useful clues about new physics if similar events are isolated and counted.
The reason P_T is important is because when the actual collision event is a short distance collision dominated by perturbative QCD, the outgoing particles are almost always away from the beam-line by a significant amount. Even in interesting events, when the outgoing particles are near the direction of the beam, it is hard to distinguish this from the much more common case of a near glancing collision, which lead to diffractive scattering.
Diffractive scattering is the dominant mechanism of proton proton scattering (or proton antiproton scattering) at high energies. The cross section for diffractive events are calculated by Regge theory, using the Pomeron trajectory. This type of physics has not been so interesting to physicists since the mid 70s, but more for political reasons. It is difficult to calculate, and has little connection with the field theory you are trying to find. But Regge theory is mathematically intimately related to string theory, and perhaps it will be back in fasion again.