I'm not an expert on imaging technology, but I'll give it a shot.
In a PET scanner you want to produce an image from the pair of 511 keV photons going in opposite directions. The reason for this is it gives you two point (detectors) to draw a line between which represent possible locations for the source. Its pretty unlikely that two 511 keV photons arrive at opposite detectors at roughly the same time if they are not from the same source/event. Therefore you can use a coincidence window to identify event pairs.
If you wanted to be fancier, you could use other methods for identifying the pairs. Light travels at roughly 1 foot per nanosecond. Thus, if you knew the exact time the photon arrived at both detectors, you could figure out where on the line between them the light originated. However, nanosecond level timing is exceptionally difficult requiring very specialized electronics and accounting for the time taken by the physical processes (such as charge collection and amplification).
If you record every event instead of just coincidence pairs, you could try to use other methods to produce an image. For example, you could include the probability that particles are attenuated by the body. These methods require MUCH more computer power, but in theory can extract more information from the scan.
FPGAs is just a fancy high speed logic/electronics kits. For example you can use more complicated logic to decide if something is from the same event. For example, you don't need to assume the detectors are exactly opposite each other (because if the photons were produced off-center the line would be a cord instead of a diameter).