This is a bit of a cheat -- it's not an insect and it lives in the bloodstream. But the filarial worm makes its way into a human host through an insect bite piercing the skin.
The life cycle of a filarial worm is interesting. A biting insect, such as a mosquito, ingests blood from an infected host. Filarial larvae enter the insects as they feed. The larvae develop in the gut of the insect. When a larva matures, it moves to the insect's head. When the insect bites another victim, the larvae travel down through the insect's mouth and into the victim's bloodstream.
Once in the bloodstream, the larvae mature into a worm that can produce more larvae capable of infecting biting insects, continuing the cycle.
There are several types of filarial worms, each affecting humans in different ways. Lymphatic filarial worms can cause elephantiasis -- extreme swelling of the limbs and other body parts.