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How does one brake a DC motor?

Essentially, you are creating a short circuit between the DC motor's contacts. The motor's shaft is still rotating when the short occurs, so it induces a current in the opposite direction than that of the applied voltage. Basically, you are using the remaining kinetic energy to switch the motor into reverse.

Try this: Take a DC motor, connect a wire between its contacts, and try to turn it. The harder you turn it, the harder it resists. Putting a variable resistor in series with this will allow you to vary how much it brakes.

The easiest way to achieve this is by connecting this wire at the relay: as soon as the relay is no longer energized, the wire (between the negative normally closed contact and the positive "in" contact) will create this short.

LEGO's outstanding MINDSTORMS Robotics Invention System is ideal for experimenting with dynamic braking.