When you jog, you run at a moderate pace, usually to get exercise. If you jog down the street, you'll pass people who are walking and be passed by people who are sprinting.
Jog can be a verb, or a noun meaning a slow-paced run: "She moved at a lazy jog around the track, in no hurry to improve her time." A jog is slower than a flat-out run. Yet another way to use the word is to mean "nudge one's memory." A courtroom attorney might hold up a piece of evidence in front of a witness and say, "Perhaps this will jog your memory."