"...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 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 after Belisarius Cawl gives it to him at the end of M41 - in a way, it is the symbol of office for the ruling monarch of the Imperium, and makes Guilliman's claim
indisputable all the more coup d'état like, since he got it from an Archmagos that innovates (seen as Tech heresy) instead of the Adeptus Custodes Captain-General or Tribunal.
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. It also has a lot of similarities with the Spear of Russ, a weapon used by Leman Russ during the Battle of Trisolian that showed similar anti-chaos abilities, managing to de-corrupt Horus for a brief moment of time. Or more relevantly, Ghal-Maraz, the signature weapon of Emps' Fantasy Counterpart Sigmar, which has shown similar anti-chaos abilities to the point where it can convert Chaos Champions back to the side of order.
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. It is worth noting that, in 2nd edition, the Sensei could receive from the Star Child a boon allowing them to spare defeated enemy characters, who would then have a slight chance to join their warband: if they were marked by Chaos, the Star Child would then purge them of the brand & most of their mutations (at least the ones divergeant from the standard human phenotype), and thus also from Chaos induced madness. And their was also a short story in the Demonhunters codex of an Inquisitor freeing himself of possession by torturing himself with his own heated power sword until the wailing demon fled his body.
In modern fluff it has no name. But the 2nd edition Fluff called it the Sword of Iron or Sword of Steel, and described it as the sword of legendary Heroes from Antiquity and Middle-Ages. As we know, the Hittite Empire used a twin-headed eagle as its emblem: they were also the first to forge weapons with iron taken from meteorites. According to Linda A. Malcor and Scott C. Littleton, that would be the prototype for the Arthurian cycle. From them also came the story of a god king - called Teshub by the Hittites - with 12 companions (representing the twelve Signs of the Zodiac; twelve is also the number of High Lords).
In the Hittite myth, Teshub vanquishes a dragon representing Chaos and is then enthroned as King of the gods and a loose Zeus analogue (this story pattern is also found in other mythologies; such as the Japanese one when Susanoo vanquishes the dragon Yamata-no-Orochi with a divine sword and finding yet another one before subsequently wed a divine princess and become King of her homeland of Izumo; or in the ancient Hebraic religion where yearly celebrations had YHWH vanquish Rahab before being crowned as King; likewise, Saint George, patron of Chivalry, rescues a royal princess by slaying a dragon. But somehow, all those European players out there who read Mechanicum just see Saint George and the modern fluff's Void Dragon. The Emperor is also represented as piercing a Dragon representing Horus and/or Chaos, using a spear and a sword, and also holding a shield with, mysteriously, a Crowned Lion instead of an Aquilla borne as heraldry (the Aquilla is here too but above the gate).
In other words, it is not so much that it doesn't have a name that it has too many, attributed through a great variety of culture (just consider all the variants for Excalibur alone: Excalibor, Escalibor, Excaliber, Caliburne, Caledfwlch, Kaledvoulc'h, Galatbrog (sword of King Nuada of the Tuatha Dé Danann), the Sword in the Stone (of Scone, likely, in the British version; and a Throne or an Altar in ancient Hittite versions) although none of them is ultimately the "real" name.
It is, simply, the Emperor's Sword.