There are several different types of detectors in use, with each one specialized to detect specific particles or measure specific quantities, so the answer varies. Look up "silicon trackers", "hadronic/electromagnetic calorimeters", and "muon detectors" for some examples.

In general, the particles radiated by the collisions register "hits" with multiple detectors, and based on the trajectory of these hits and the energy deposited at each one, you can figure out the properties of the particle that made them (e.g. charge, momentum).