It is a turn counter. It means that the enemy has a special ability (usually nasty) that is activated on the turn that starts with one (1) on the counter.

The counter decreases by one on every turn that the enemy is able to focus. You can break enemy focus by making them fall which stuns them. Non-sentient enemies (e.g. bombs) cannot be stunned and they will count down even when falling.

To find out the target's special ability, tap and hold on the target.