What is it?
Nail a Big Daddy with this gooey green orb and you'll get what the name promises - a brainwashed bodyguard who believes you're his Little Sister and who will pound anyone who even tries to mess with you.
When should I use it?
If you have the plasmid handy and a Big Daddy nearby, there's no reason not to use it. Fighting alongside BioShock's iconic bad boy ranks among the coolest moments of the entire game. Not only will he assail your enemies, he'll follow you from room to room and sacrifice himself to protect you, putting his own body in the Splicers' line of fire. Awesome beyond words.
A note of caution: Do not hurt the Big Daddy while he is on your side. Hitting him with a single attack... even accidentally, even splash damage... will immediately turn him against you. You'll know you've made a mistake when the happy green glow inside his helmet transforms to stop sign red. For that reason, you should never, never use Insect Swarm in these situations. The bees' attack patterns are just too unpredictable.
In fact, now that we think it over, maybe you should just cower in a corner like the Target Dummy when under the Big Daddy's watch.
Does it work against a Big Daddy?
Well, obviously it works on a Big Daddy, but does it work against one? Sort of.
As we just mentioned, you can't attack him - let alone kill him - while he is your guard. And, as you can't directly command him, it would be very tough to have him walk into any kind of trap. Your best bet is to have him fight as many Splicers as possible - eventually, he'll be taken down. If not, at least you obliterated every other enemy in the level. Also be sure to keep reapplying the Hypnotize plasmid... it does wear off.