Those particles appear if you have a summoned minion (such as the Imp or the Hornet) and you have right-clicked on the monster while holding the summoning tool. Three small purple particles will revolve around the monster you clicked, and your minions will try to focus on that target instead of any other nearby enemies. This is useful in boss fights where the boss summons a bunch of smaller monsters (such as the Eye of Cthulhu or the Wall of Flesh), since it increases your DPS against the biggest threat.
This is a new feature in 1.3.4 - prior to this version, minions would simply attack whatever enemies were closest/easiest to reach.
Edit:Further info: While playing around with the summoning items, I did notice that often you don't even have to click directly on a monster - it's quite possible that you had the summoning staff equipped and right-clicked in the general vicinity of the monster. This would still attach those particles if the monster was "close enough" to your cursor when you clicked.