Schematic of Anakin's Second Lightsaber (and Luke's first) from Star Wars: The Visual Dictionary.
Luke is shown constructing his second lightsaber before going to Jabba's palace; the images bear a striking resemblance to the deleted scene from Return of the Jedi (video included below, near the end of this answer).
Star Wars Databank:
An essential rite of passage for young Jedi is the Gathering. In the Gathering, younglings go to the frigid ice world Ilum, where they search for kyber crystals to build their first lightsabers. The crystals are located in the planet’s Crystal Cave, an intimidating maze, and “call out” to the Jedi they are in tune with. Once the correct crystal has been found, younglings rendezvous with Huyang, and ancient droid who chooses the parts necessary to build a lightsaber. With everything in place, younglings complete the ritual by constructing their Jedi weapons through a graceful use of the Force.
Huyang was an ancient droid, stooped with age yet still spry of mind. Huyang was a storied architect and professor, who oversaw the construction of lighstabers for centuries. Based out of the training vessel Crucible, Huyang guided younglings after they had successfully completed their task of securing a kyber crystal from the icy caverns of Ilum. Huyang helped them visualize their lightsaber parts, and pulled the needed supplies from his vast trove. His ancient memory systems contained records of every lightsaber constructed.
Star Wars Databank has a video of Huyang instructing Younglings in lightsaber construction, from The Clone Wars.
At the heart of every Jedi lightsaber is a kyber crystal found on several planets, most notably the icebound caves of Ilum. This crystal is attuned to the Force, and connected to a Jedi Knight on a deeply personal level. In this way, a lightsaber is an extension of a Jedi's Force awareness. Because Jedi let the Force guide their selection of the crystal, the vibration that the crystal creates in the lightsaber blade helps Jedi center themselves and find balance in the Force. In this way, a Jedi can center his or her attention beyond the distractions of combat. A lightsaber crystal is colorless until first attuned and connected to a Jedi -- at which times it glows either blue or green, or in some rare instances, another shade. From that point on, it retains that hue.
The Databank also has a clip from The Clone Wars in which Yoda prepares the Younglings to harvest the crystals for their lightsabers.
The most sacred world to the Jedi Order, the ancient planet Ilum is forever encased in ice, and whipped by biting winds. Hidden beneath the frozen surface is a Jedi Temple from the very dawn of the Order. It is here, within the legendary Crystal Caves, that Jedi initiates must undergo The Gathering. The rite of passage, handed down over centuries, requires Jedi younglings to trust their instincts, overcome inner obstacles, and find the unique Ilum crystal that is attuned to their Force presence. Upon retrieving the crystal, a Jedi initiate begins the process of constructing his or her lightsaber, under the guidance of Professor Huyang.
Star Wars - The Clone Wars:
This article has a clip of Younglings gathering crystals.
Star Wars - Heir to the Jedi
Luke dismantles a lightsaber:
My room had a basic desk and chair, and I seated myself at it and placed the lightsaber on the desk, emitter pointed away from me for safety. As before, I kept myself open to the Force, but now I tried to focus on the lightsaber and feel the Force inherent in it. Closing my eyes, I explored the top of the hilt right below the emitter with my fingers, searching for any tactile clues. The surface retained that same strange slick feeling, but I detected nothing unusual at the top, or around the button or dial, or even on the rest of the hilt. When I ran my finger fully around the base, however, clockwise and then counterclockwise, eyes still closed and trying to feel the Force, a snick announced the appearance of a fissure lengthwise down the hilt; after another soft click, the casing popped free, revealing yet another metal sheath, one that looked more like mine and had visible screws. Artoo unscrewed them for me and I was able to lift off one half of the sheath and reveal the innards.
The power cell at the base was insulated and held no interest for me. Above that was a platform for the primary focusing crystal that gave the lightsaber its color. Two additional crystals floated above it, balanced so precariously on mounting ridges that they could easily be disturbed—and they had been. They lay askew, and I feared I must have done that in the process of disassembling it. The lightsaber wouldn’t work properly now, even if I put it back together; without proper focusing there was no telling what would happen if I tried to turn it on. It might explode. And aligning those crystals by hand would be impossible—I sensed that it had to be done with the Force, and only through the Force would I know whether it was aligned properly or not. They were wafer-thin slices of crystal, too, a beautiful clear amethyst, and might scratch or cloud with handling. Moving them precisely with the Force would ensure that they remained pristine.
The lightsaber’s construction confirmed for me what I had already suspected: Far from being merely a feeling of interconnectedness that could guide your actions or a method of tricking the weak-willed, the Force could be used to manipulate solid objects. However, the skill required to construct a lightsaber—or even put this one back together—was a parsec or five beyond my current abilities.
I had Artoo take holo stills of the lightsaber as I deconstructed the rest of it for future study, and then I thought I should work on those Force abilities if I ever wanted to reassemble it or make my own.
Star Wars - Return of the Jedi (novelization):
But this lightsaber Luke had fashioned himself, in Obi-Wan Kenobi’s abandoned hut on the other side of Tatooine — made with the old Master Jedi’s tools and parts, made with love and craft and dire need. He wielded it now as if it were fused to his hand; as if it were an extension of his own arm. This lightsaber, truly, was Luke’s.
Star Wars - Rebels:
Luke constructing his lightsaber - Star Wars Galaxy trading cards, by Frank Stockton
Star Wars: The Card Game – Balance of the Force (Card: The Force Is Strong) by Anthony Foti
Luke constructing his lightsaber - "Knight of Passage" by Jerry Vanderstelt
Anakin Skywalker constructs his first lightsaber by Chris Trevas
Star Wars - The Old Republic:
Star Wars - Shadows of the Empire (novel - once canon, now Legends)
Luke builds his second lightsaber - the green one we see in Return of the Jedi - over months, and the scenes in which he works on the saber are scattered throughout the book:
He no longer had a lightsaber, though he’d started constructing another one, using the plans he’d found in an old leather-bound book at Ben Kenobi’s. It was a traditional exercise for a Jedi, so he’d been told. It had given him something to do while his new hand had finished final bonding to his arm. It had kept him from thinking too much...
Luke stared at the little furnace, as if so doing could hurry the process. Inside, the ingredients for a lightsaber gem cooked at an incredible heat and pressure, hot enough to melt denscris, intense enough to collapse durasteel into a liquid ball...
His teacher hadn’t left much behind, not for a man who had once been Obi-Wan Kenobi, a Jedi Knight and a general in the Clone Wars. Perhaps the most valuable thing was the old and intricately carved boa-wood trunk and its contents, including an ancient leather-bound book. A book that contained all manner of wondrous things for a would-be Jedi, such as plans for building a lightsaber.
According to that book, the best lightsabers used natural jewels, but there weren’t a lot of the kind he needed lying around where he could find them on Tatooine. He’d managed to collect most of the electronic and mechanical parts in Mos Eisley— power cells, controls, a high-energy reflector cup— but he had to make his own focusing jewel. Ideally, the best lightsabers also had three of those, different densities and facets, for a fully adjustable blade, but for his first attempt at building the Jedi weapon, Luke wanted to keep it as simple as possible. Even so, it was trickier than the book made it out.
He was pretty sure he had the superconductor tuned right, the amplitude for the length set where it was supposed to be, and the control circuitry boards correctly installed. He couldn’t be positive until the jewel was finished, and the book didn’t mention exactly how long that took. Supposedly the furnace would shut down automatically when it was done. If everything went right, he’d be able to cut the jewel, polish and install it, tune the photoharmonics, and then he’d only have to hit the switch to have a working lightsaber.
So far he’d been very careful constructing the thing, triple-checking each step, and to get this far had taken almost a month. The book said a Jedi Master in a hurry could construct a new lightsaber in a couple of days...
But maybe he would just finish the facets on the lightsaber jewel first. He had a lot to think about, and he didn’t think he would be able to take a nap just yet; too many things buzzing around in his head. Might as well do something useful. He put on a robe and moved to the worktable...
Luke took a deep breath. He stood outside Ben’s house, the first stars of evening aglimmer, the moon still on the rise. The air was warm but not as scorchingly hot as it had been. He held the completed lightsaber in his right hand. He had assembled it according to the old book’s direction; everything should work.
Should work. But he’d come outside to test it. That way, if it blew up, at least it wouldn’t take Ben’s house with it.
Artoo stood nearby, watching. Luke could have had the droid try it without any risk to himself, but what kind of Jedi would do that?
“Go back inside,” he told Artoo.
Artoo was not happy with that and said so, ending in an air-forced-through-rubbery-lips noise.
“Go on. If something happens, I need you to tell Leia.”
Yeah. Tell her Luke, the galaxy’s biggest idiot, flash-flamed himself into a black crisp because he couldn’t follow an elementary circuit diagram.
Artoo left, whistling his protest as he went.
Luke let his breath out. He waited until Artoo was out of sight, then took another deep breath, held it, and pushed the control button—
The lightsaber glowed; the blade extruded to full length, just under a meter, and began to hum with power. It gave off a green gleam that was quite bright in the early night.
Mashable article titled A Jedi's Weapon - The History, Construction and Lore Behind the Lightsaber:
To Jedi knights, the construction of a lightsaber is an important step upon the road to becoming a warrior.
Finding and selecting the rare crystal at the heart of the weapon is a rite of passage for young Jedi students. Building the tool itself requires care, as a poorly constructed weapon can fatally backfire.
You won't hear about crystals or caves in the six existing Star Wars films, and don't expect a thorough breakdown of lightsaber "science" in Episode VII. Much of the lightsaber mythos comes from the vast, expanded universe of books and games -- though recent animated series like The Clone Wars and Star Wars: Rebels have added to the canon.
Deleted scene from Return of the Jedi:
Star Wars Canon Catch-Up: What Are Lightsaber Crystals?
Story of the Lightsaber: Lightsaber Past, Present and Colors
How to Build a Lightsaber
A Jedi's weapon: The history, construction and lore behind the lightsaber