FSD supercharging is the process of flying into a high-energy relativistic jet of a neutron star or a white dwarf in supercruise, which increases the FSD range for the next jump at the cost of slight damage to the Frame Shift Drive itself. FSD supercharging from a white dwarf gives a 50% range bonus, whilst doing the same with a neutron star provides a 300% bonus, effectively quadrupling your range for a single jump. Note that supercharging is not an instant process, and the pilots may have to spend several seconds inside the beam with a Fuel Scoop for the effect to apply.
However this is a risky process, as not only would your ship's controls be disrupted by the high-energy beam of the star, but disengaging supercruise while inside the jet will result in the entire ship experiencing major strain to all its modules and the hull itself. So far it seems like bigger ships such as Asp Explorer or Anaconda are much more resistant to those effects than smaller ones, e.g. Eagle.
By using a Neutron Highway you can travel large distances with far fewer jumps than the conventional method.
Double Supercharging Edit
[already nerfed] It was possible to gain a second FSD supercharge, while leaving the system. This effectively allowed a commander to use the supercharge, upon arrival at the next system, even if there is no neutron star present. This is less risky, but more difficult. It required a pilot to line up the next system destination, so that it can be seen down the cone of the energy jet, facing away from the star. The pilot was required to charge their FSD drive, as they slowly approach the blue color of the jet, then throttle up, right before entering. This would thrust the ship into hyperdrive, while travelling down the cone to your destination. As you leave the system, your original charge will be used, thus allowing the second charge to take place.