Aircraft are usually though not always hired externally as it is cheaper to do this than for an aid agency to maintain it's own fleet.
Aircraft range from Pilatus Porter to DC8 ,though Islanders ,Twotters ,Caravans feature pretty heavily.
Can be dangerous depending on where you end up ,yes aircraft have been shot down ,hijacked ,crew held for randsome and some countries are littered with land mines.
Some good jobs out there some very dodgy ones as well.Best to work for someone big like Red Cross or United Nations , with back up if things go wrong.
Try a searh for Zimex Aviation and Ross Air ,both heavily involved in relief flights.