The beginner definition is useful: in man coverage, defenders are primarily responsible for particular receivers. In zone coverage, defenders are primarily responsible for areas of the field.
The complication is that modern defences use match principles that begin like zone and turn into man based on the routes. So the goal is not to identify everything perfectly before the snap. It is to collect clues.
Clue 1: motion
If an offensive player motions across the formation and a defender follows him all the way, that is a strong clue for man coverage. It is not proof: defences can deliberately “travel” a defender or disguise their coverage.
If the defence bumps responsibilities across the formation rather than sending one defender with the motion player, zone becomes more likely.
Clue 2: cornerback leverage
A corner lined up directly across from a receiver with inside or outside leverage may be preparing to play man. A corner sitting farther off and looking into the backfield may be more likely to be responsible for a zone.
Clue 3: eyes after the snap
This is often the best clue. Man defenders tend to turn and run with receivers. Zone defenders are more likely to keep their eyes on the quarterback while passing receivers from one zone to another.
How offences attack man
Man coverage creates one-on-one matchups, so offences use crossing routes, bunch formations, motion and picks or “rub” concepts to create separation. Mobile quarterbacks can also become dangerous because defenders have their backs turned while chasing receivers.
How offences attack zone
Zone defence creates spaces between defenders. Offences try to stretch those zones horizontally or vertically and place receivers into the gaps.
Against Cover 2, for example, a quarterback may attack between an underneath corner and a deep safety. Against Cover 3, the seams and underneath hook areas become important.
Best fan clue: watch defenders' eyes after the snap. If they are staring at receivers and running with them, think man. If they are staring at the quarterback and exchanging receivers, think zone or match-zone.