Who invented the lightsaber?

Question

As we all know, in every Star Wars movie we can see all the Jedi using various colors of lightsabers to fight.

Do any of the movies mention who invented the lightsaber and/or why the Jedi started using them?

Answer 1

The answer to this question depends if you are talking about Legends (pre-Disney) or canon (post-Disney).

Legends:
If you consider Force-sabers (predecessor of the lightsaber) then the first creators where the Rakata of the Infinite Empire who were able to channel dark side energy through synthetic crystals to form a saber like weapon.

Otherwise the first lightsabers were made by the Je'daii Order on the planet Tython made by a smith only known as "Weapon Master" (2011 Star Wars: The Old Republic) made by combining technology from different worlds to "freeze" a blaster beam.

15500 years before the first Death Star got destroyed the blades became an integral part of the Jedi Order but in a different form, these original versions required belts attached to the blade to power them. These blades were mostly ceremonial, and it was the Sith adapted them to be able to be used without belts and make them stronger. It was only 5000 years before the Death Star’s destruction that Great Hyperspace War started in which the Sith and Jedi fought again bringing these Sith advancements to the Jedi.

But this is no longer considered canon now Mickey is the Emperor so the canon answer is:
They predate the (First) Republic by thousands of years, being used by the Jedi/Sith during their war on Malachor (According 2015 Star Wars: The Force Awakens: The Visual Dictionary).

So these are the origins of the lightsabers in both versions, none of the films touch on this subject and the information can only be pieced together from books/games.