Surprisingly, in Star Wars a non-Jedi can develop a Force bond with a lightsaber as well. Many Jedi Knights see the forging of their own lightsaber as a key moment in their personal growth. Every Jedi chooses to make their lightsaber out of components that are precious to them in some way; Rey's new yellow lightsaber in Star Wars: The Rise of Skywalker, for example, incorporated parts from her staff.
At the heart of a lightsaber is a kyber crystal. These living crystals are incredibly rare, and they are perfectly attuned to the Force. The kyber crystal bonds with a Jedi even as they construct their own lightsaber, and as a symbol of this bond the crystal changes color; this lends each lightsaber blade its color, based on the kyber. It's important to understand, though, that a lightsaber is no mere weapon; as noted in Chuck Wendig's tie-in novel Aftermath: Empire's End, it serves as a tool to aid the Jedi's focus. "Just as the Jedi are a lens that focuses the Force, so is the kyber crystal a lens that focuses the light inside the Jedi - and the light inside the Jedi's weapon, the lightsaber," a leader in the Church of the Force explained in that book.
Surprisingly, a person does not have to be a Force-sensitive to bond with a lightsaber. This is because kyber crystals potentially allow a non-Force-sensitive a limited degree of access to the Force, as revealed in Rogue One: A Star Wars Story. In one key scene, Jyn Erso led a group of Rebels to the planet Scarif, and they awaited clearance to slip into the Imperial world undetected. As the tension built, Jyn clutched the necklace she had been given years ago by her mother - one that held a kyber crystal within it. A moment later, the Imperial on the other end of the comm channel gave the Rebels clearance. Although it wasn't overtly stated, the scene's implication was that Jyn had somehow touched the Force. In other scenes, Chirrut Imwe - one of the Guardians of the Whills - demonstrated preternatural powers of his own. Lucasfilm clarified that Chirrut was not Force-sensitive; but, significantly, both his staff and his lightbow contained kyber crystals.
Star Wars Rebels developed this idea when Sabine was given the Darksaber, and Kanan began to train her. Although Sabine was not Force-sensitive, Kanan trained her just as he would a prospective Jedi Apprentice, encouraging her to bond with the blade until she became one with her lightsaber. His guidance paralleled observations from Luke Skywalker himself in the recently-published Secrets of the Jedi book:
"It's said that wielding a lightsaber is less like swinging a sword and more like directing a current of power. When you're attuned to the Force, your thoughts and actions all become part of the same flow of energy, which is then directed through the kyber crystal and into your lightsaber's blade."
Of course, no non-Jedi will be able to use the Force quite so creatively as a Force-sensitive. Indeed, they shouldn't even be able to swat aside blaster bolts; that requires a far more significant awareness of the Force, a talent called "double vision" in which the Jedi experiences both present reality and future threat at the same time. A Jedi has trained to trust the Force on an instinctive level, and thus moves the blade to block the blaster bolt, potentially even returning the shot. Jedi Apprentices learn to block blaster bolts early on in their training, precisely because the skill develops that deep and intimate relationship with the Force. In Star Wars Rebels, Kanan notably stopped training Sabine well before he got to that point, suggesting he did not expect her to be able to achieve this even when she had bonded with the Darksaber.