PET is based on the fact that the positron won't travel very far before undergoing annihilation. When annihilation happens, two gamma photons are emitted, moving in opposite directions. Detectors are arranged in a ring around the patient. The scanner registers pairs of photons that reach different parts of the detector ring near-simultaneously and localizes the annihilation events based on these. Since positrons only travel very short distances (<1 mm) before annihilation, the distribution of annihilation events corresponds to the distribution of the radiotracer molecule within the body.
At least that is the med school explanation, I may not be qualified to talk about the finer points of the physics behind PET.
Yes, that's the idea. PET works by placing biologically active molecules (like glucose) that have been "tagged" to contain positron emitting isotopes into the body. The body then takes up those molecules as part of its normal operations and you can watch how the spread. The positrons emitted from the molecules annihilate a short distance away and produce highly penetrating gamma rays that can be picked up by a detector. The end result is to basically place "glow in the dark" materials in your body except they glow in gamma rays. The PET scanner crunches the data on where the gamma rays are detected from to deduce what the light source (the 3D structure of different concentrations of the gamma-ray "glow in the dark" tagged molecules) looks like inside your body. This can be used to determine the structure and function of various metabolically active parts of the body (such as organs or even tumors).