Hit Points, short for HP, is the amount of damage needed to clear a certain stage. The pink bar below the target Pokémon icon shows the progress, beginning at full bar. When the bar is completely depleted, the player has cleared the stage. Competitive Stages do not have an HP bar, since the goal is to simply rack up as much damage as possible. Meowth's Coin Mania also does not have an HP bar.
Some disruption countdowns will switch upon crossing certain HP thresholds.