The sword of Turgon and Gandalf.
"This, Gandalf, was Glamdring, Foe-hammer that the king of Gondolin once wore."
(Words of Elrond, from The Hobbit 3, A Short Rest)
Of Glamdring's early history, we know almost nothing. From Elrond's words in 'The Hobbit' ('This, Gandalf, was Glamdring, Foe-hammer that the king of Gondolin once wore'), we can say that it was forged some time in the second to fifth centuries of the First Age (the centuries when Gondolin existed), and that it was borne by Turgon (the only Elf to hold the title 'King of Gondolin'). After Gondolin was built, Turgon was in battle twice: at the Nirnaeth Arnoediad and at the Fall of Gondolin. Glamdring was greatly feared by the Orcs even thousands of years later, so Turgon must have wielded it ferociously on both these occasions.
After Gondolin's Fall, Glamdring passed out of history or rumour for more than 6,000 years. The fact that it survived the War of Wrath, though, tells us that it must have left Beleriand before the end of the First Age, perhaps carried eastward by a band of Orcs. More than that cannot be said, except that somehow, by the late Third Age, it had fallen into the hands of the three Trolls called in The Hobbit Tom, Bert and Bill Huggins.
It was discovered in their troll-hoard by Gandalf, Bilbo and the Dwarves as they travelled eastward on the Quest of Erebor, and claimed by Gandalf. After it was identified by Elrond, Gandalf wore if for eighty years. He had it with him when the White Council drove Sauron from Dol Guldur, and he bore it through the War of the Ring itself.
Famously, he used it to battle the Balrog at the Bridge of Khazad-dÃ»m, and it fell with him there into the abyss. Glamdring survived that fall: Gandalf had it with him when he returned, and he bore it throughout the last days of the War of the Ring and beyond. We know that he still had the sword when he rode through Bree with the Travellers in October III 3019, but Tolkien doesn't mention it after that point. It seems safe to assume, though, that Gandalf was still wearing Glamdring, as he had for eighty years, when he boarded the White Ship at the end of the Third Age.
We know much less about Glamdring's appearance than its history. It is consistently described as 'white' (in The Fellowship of the Ring II 5 The Bridge of Khazad-dÃ»m). When enemies were near it 'shone with a pale light' (ibid). In the same circumstances, the Elven-knife Sting glittered along its edges, but in Glamdring's case, the entire blade seems to have glowed. Finally, we know from The Hobbit that both Glamdring and its mate Orcrist had runed blades, giving at least their names.