It took me quite a few modified swallow studies with the Speech Therapist in order to visually witness what they were looking for. This clip (played three times) shows exactly what they are looking for:
This video shows aspiration - I allowed it to play twice before adding an arrow pointing out the aspiration on the third loop. Watch the patient finish chewing... observe the black bolus travel from the oropharynx, down to the laryngopharynx, then the epiglottis closes off the trachea. The majority of the bolus travels down the esophagus (parallel to the spine, but posterior to the trachea). The arrow indicates a small amount of contrast getting around the epiglottis and dripping down the trachea. Notice that the patient does not seem to have a cough reflex, which is not quite as common.