Edge triggering happens when some stimulus changes; level-triggered things happen when some stimulus crosses a threshold. In the case of digital circuits, the stimulus is voltage on a wire -- edge-triggered things happen when the voltage changes, level-triggered things happen when the voltage is above or below some pre-set amount.
As an example, consider your telephone. It rings when someone dials your phone number (edge-triggered) and has an indicator to let you know when there's at least one message waiting (level-triggered).
As an example, consider your telephone. It rings when someone dials your phone number (edge-triggered) and has an indicator to let you know when there's at least one message waiting (level-triggered).