Taking Damage is what happens when you are hit by an enemie's attack. Most normal enemies attacks will cost you 1 or 2 damage. However, some bosses' attacks will cost you more than 2 damage. Damage is represented by the loss of a heart, and you have 5 Hearts to start with at the beginning of each level. If you take damage, a heart can be regained, as hearts can often be found throughout levels, and sometimes killing an enemy will drop a heart.
Also see: Heart