It was an emotional moment when Jaime Lannister bequeathed his Valyrian steel sword, Oathkeeper, to Brienne of Tarth. Quite aside from the Jaime/Brienne 'shipping, Valyrian steel is pretty damn rare in Westeros in the time in which the show is set. Like so many things, there are a few differences between which Valyrian steel swords remain in the books as opposed to in the show, so I'll be sticking to the show for this article to keep things simple.
How Many Valyrian Steel Swords are Left in Westeros?
Made when Tywin Lannister ordered the melting down of Ned Stark's greatsword Ice. Given to Jaime Lannister, who gifted it to Brienne of Tarth.
Owner: Brienne of Tarth
2. Widow's Wail
The second sword made from Lord Stark's greatsword, Ice. Given as a gift to Joffrey and passed to his brother Tommen on his death. Now li'l baby Tommen is also dead, Widow's Wail's owner is uncertain.
Owner: House Lannister
The ancient sword of House Mormont, given to Jon Snow by Jeor Mormont of the Night's Watch.
Owner: Jon Snow
House Tarly's ancestral sword, stolen from Randyll Tarly of Horn Hill by his son.
Owner: Samwell Tarly
Lost Valyrian Steel Swords Mentioned in the Show
5 & 6. Blackfyre & Dark Sister
The ancient swords of House Targaryen. Lost in the Blackfyre rebellion.
House Lannister's valued family heirloom, lost on an expedition to Valyria hundreds of years before the action in the Game of Thrones show. The loss of the Lannister's Valyrian steel was what prompted Tywin Lannister to melt down Ice to create great Lannister swords anew (and obviously, it was a great poke in the eye to the Starks, too!).
Bonus: The dagger used in the assassination attempt against Bran Stark had a Valyrian steel blade and dragonbone handle. It belonged to Petyr Baelish, who says he handed it over to Tyrion Lannister when he lost a bet.