Emperor's Sword
"...when the Evil Spirit assailed the creation of Good Truth, Good Thought and Fire intervened."
- – Yasht 13.77, from a collection of 21 Zoroastrian hymns
The Emperor's Sword was used by Himself during the Great Crusade and by Primarch Roboute Guilliman as of the 42nd millennium.
The Emperor's Sword is a relic of unknown origin that dates back to at least the Great Crusade. It has some super-special-powers that are unique to it (see below); and is, almost as importantly as of 8th Edition, a plot device to show that Guilliman is the rightful commander of the Imperium - in a way, it is the symbol of office for the ruling monarch of the Imperium, and makes Guilliman's claim indisputable like a grimdark Excalibur or what ever version of the story your reading calls the actual sword in the stone.
Obviously, this sword has a lot of meaning and importance for not just the Imperium at large, but also for many special characters, in particular in the books. When Dante is having prophetic visions during Devastation of Baal, he sees the Emperor in His throne without the sword, which freaks him out immensely, and kept pondering upon the implications of that vision for a while. This was resolved once the battle ended, after seeing Guilliman wielding the sword, and realizing the visions did not mean catastrophe but instead one of hope, the return of a Primarch. In Dark Imperium, Guilliman also reflects about the nature of the sword, and whether its properties are of divine or mundane nature, seeing as they are capable of banishing daemons even he would've struggled with if he hadn't had Daddy's sword. He notices that it is in fact a Force Weapon but somehow Guilliman doesn't need psychic tests to activate it or perhaps he counts as a psyker and can therefore use it in the presence of Sisters of Silence & such as if they were not here, nor be troubled by their unpure presences. It also changed size to fit in his hand, but he can't remember what size it and the Emperor was before. Guilliman also notes there are secrets to the weapon that only the Emperor knows how to unleash. This, combined with his view on the Emperor's humanity-divinity (on top of all the shit he has to worry about in the book), makes him quite stressed out and melancholic of better times at the end of the book.
According to the sources, the blade is not only a powerful weapon in its own right, but it also has some of the most powerful cleansing abilities seen to date. Capable of dispelling Chaos energies and banishing daemons with ease, in point of fact, it can permakill them, annihilating their souls so they are no more, forever. This blade was probably made by the Emperor to specifically fight the malevolent forces of the Warp, and probably saw a lot of use during the War in the Webway, during the Horus Heresy. Considering how big of a McGuffin the Emperor's Sword is being in recent books, it's pretty safe to say Black Library probably has something prepared for it in the not-so-distant future.
In the 2nd edition Fluff it's called the Sword of Iron or Sword of Steel, and described it as the sword of legendary Heroes from Antiquity and Middle-Ages. Since these heroes were invented around the same time, it couldn't have been used by all of them due to overlap. It's also too fucking big for anyone smaller than an Astartes Terminator to weild. So either it was created from their remains or the Emperor may have actually been about half those heroes. Storing it like the Master Sword from Zelda when he didn't need it. Humming or playing his own item get theme when he pulled the sword out of its pedestal. But that has been Retconned.
It has no name, it needs no name.
It is the Emperor's Sword.
No word on what happened to his lightning claw though.