The lightning rod is based on two principles theorized by Benjamin Franklin. Lightning dissipation theory, and lightning diversion theory.
Lightning Dissipation Theory
This theory says that if you point a pointy metal object toward a polarized cloud, the metal object will be able to bleed off some of the energy from the cloud. Thus preventing a lightning strike.
This theory can actually be demonstrated, using a Van de Graaff generator and a nail. This YouTube video demonstrates the theory.
While this theory holds up on the small scale, it's been shown not to be effective at dissipating the large amount of energy built up in a storm. Fortunately, the design of the dissipation device (lightning rod) is also a great diversion device.
Lightning Diversion Theory
The lightning diversion theory says that if you provide a preferable path for the energy to travel along, there's a high probability the energy will follow that path.
Lightning rods are designed to be the highest objects around. This puts them closer to the polarized cloud, and reduces the distance the lightning must travel through the air. They are also made from conductive materials, and are connected to the earth through highly conductive materials. This provides a low resistance path to ground, making it a preferable path for lightning to follow.
While both theories hold up in the laboratory, only diversion theory seems to offer a viable lightning protection system.
Best Answer
The conventional definition of lightning is a current though a plasma (not necessarily through air as lightning happens on other planets) so in a vacuum there cannot be any lightning. However charge does still flow between electrodes in a vacuum, and from personal experience I know that we can get something very like lightning in the right circumstances.
In a vacuum charge can leave a surface due to field emission. Basically the field gets so strong that the electrons at the surface can get enough energy to overcome the metal work function by moving only a small distance and they tunnel out of the metal. However this is a rather diffuse flow and wouldn't generate any light so it wouldn't resemble lightning.
I did claim you can get an effect like lightning, and this can happen if there is a rough spot on the electrode surface that concentrates the electron flow. If the electron flow gets concentrated enough it can vaporise the electrode at the high point then you get an arc through the vapour. In the old days when we used giant Van de Graaff generators as particle accelerators this used to happen on a regular basis leaving a scar on the dome that then needed to be polished away.