Elvish Sword names
I don't recall that it's ever mentioned. Elvish swords don't seem to have had much to do with the Fellowship's Quest. Elvish bows and arrows did: Galadriel gave Legolas a very fine bow to replace his small one from home, and he used it to excellent effect. There was some mention of Elvish knives, I believe, though only a little. Sting (which wasn't a sword at all, but a long knife), Orcrist (Goblin-Cleaver, which passed to Thorin), and Glamdring (the Foe-Hammer, which passed to Gandalf) appear to have been Elvish weapons, having been forged ages ago in the hidden Elvish city of Gondolin ("hidden" because it was deeply nested in a basin of mountains, practically inaccessible - even its name meant "Stone-Hill place") for what Bilbo simply referred to as the Goblin Wars. Gondolin was in the formerly vast region west of what became the Shire, called Beleriand, which was destroyed - half a continent of it, sunk below the sea - when the Valar brought their full power against Morgoth. So these weapons are among the few artefacts remaining from Gondolin. Little or nothing seems to be said about the style or type of them, except that they presumably weren't scimitars - Orcs used those. They had extremely good edges and were very strong - Bilbo drove Sting into a wooden beam just to demonstrate to Frodo how easy it was - and a bluish gleam or fire would dance up and down their edges with the proximity of Orcs. Frodo was able to stab Sting into the foot of a great cave troll when it had already just turned Boromir's sword aside. If anything similar was used by Elves of Rivendell or Lorién or Mirkwood, I don't recall any mention of it.
If there was any inscription on Glamdring or Orcrist, I don't remember it.
The swords of the Dúnedain (and of the Orcs) and the axes of the Dwarfs were more notable. Though even there, the main one was the ancient sword Narsil, which had belonged to Isildur but broke beneath him in combat, and which Elrond's smiths reforged for his heir Aragorn, who renamed it Andúril (Flame of the West). I believe King Thëoden's sword was mentioned by name, but I don't remember it, and what kind it was was never brought up; neither was the kind of the sword that Denethor wore under his robes, even in the throne room of Minas Tirith.
And of course the other hobbits had large knives, not swords, like Sting. They apparently weren't Elvish, having been created by the northern Dúnedain in the wars against Angband (in which the Men were defeated and scattered, leaving such weapons and other artefacts in cairns and barrows etc), but had Elvish influence enough to have been crafted with runes or other inscriptions that conveyed power against Morgul.
An idea seems to have grown up that weaponry and fighting skills should have been important to the Fellowship, and I suspect it's this that has sparked a lot of wondering about Middle-earth swords. Actually, it was very important to them to avoid encountering any enemies while carrying the Ring. Even if the enemies had no idea about the Ring, which was almost certain, it would have been utter disaster if they'd been able to seize Frodo, even just his body. Stealth and the avoidance of spies were essential to them, which partly accounts for the discomforts of their journey while still west of the Misty Mountains. Of course, there were times when evasion was impossible, and then the fighting skills of bigger members of the party became very important; but for the most part they tried to keep Frodo well away from any enemies.