This is a green lacewing adult on a milkweed plant covered with milkweed aphids (click on any of the photos below for a larger image).  When the adult finds a good food source for its larvae...

...it lays its eggs on the tip of a very fine strand.  This is because the larvae are such voracious eaters that if the eggs were all together, the larvae would eat each other once they hatched.

This is a green lacewing larva after it's hatched.  It will devour aphids and other garden pests.  Since there are far more aphids than lacewings, it's likely that using pesticides would kill ALL the lacewings (and other beneficial insects), but only kill SOME of the aphids.  So the aphids would come back in full force without any natural predators.  Nature has a way of taking care of itself.

CORRECTION 8/12/07.  I originally identified the larva in the picture below as a Green Lacewing larva, but I now believe it's the larva of a Brown Lacewing.  They're very similar, but the Brown Lacewings lay eggs directly on the plant, not on thin strands like Green Lacewings do.  Brown Lacewings are equally beneficial in the garden.  There's a photo of the adult below the larva photo.