Lightning Fragments are items nessisary to the crafting of Plasma. They are only obtained from a lightning bolt striking a Lightning Catcher. They can only be stored in Thunderchests, otherwise a regular chest will explode.
Lightning Fragments can also be used for fighting. If you right-click the ground, it will have an electric field surround a 2x2 radius that shocks and paralizes mobs that walk in it. You can tell when a mob is paralized when it is not moving and a yellow sparky aura surrounds it. If a Lightning Fragment is in your Hotbar, whenever you hit a mob (it does not matter what the weapon/item is that you use to hit an entity with), there is a varied chance that the mob will be shocked and become paralized for a short time. The more Lightning Fragments, the greater the chance to shock. Every Lightning Fragment in your hotbar increases the shock chance by 1%, but the highest chance you can get is 64%. Extra stacks will not affect the maximum amount. If the Lightning Fragments are in your inventory, they increase the chance by 0.5%, but cannot help to exceed the limit of 64%.