Valyrian steel

Valyrian steel is a form of metal that was forged in the days of the mighty Valyrian Freehold. It is exceptionally sharp, tremendously strong, and extravagantly expensive.

Some maesters also bear a valyrian steel link in the maester chain they wear. It is a sign that said maester has studied the "higher mysteries" - magic. This field of study, however, is mostly theoretical and its purpose is to demonstrate that magic, if it ever existed, is now extinct.

Valyrian steel swords

 * Dark Sister, a longsword of House Targaryen, held by Visenya Targaryen, now lost.
 * Ice, the greatsword of House Stark, now in the possession of Ser Ilyn Payne.
 * Longclaw, a bastard-sword of House Mormont, given by Jeor Mormont to Jon Snow.

In the books
In A Song of Ice and Fire novels, the secret of forging Valyrian steel was lost in the Doom, making weapons made of it extremely expensive. Skilled smiths can reforge Valyrian steel weapons by melting down existing ones, but it's a difficult process. The master-blacksmiths of Qohor are noted as being among the few who can successfully reforge it.

Although Valyrian steel blades are scarce and costly, thousands of them are known to exist in the world, approximately two hundred in Westeros alone. Most of them are swords, but there are a few daggers and axes as well. Valyrian steel can be identified by its dusky, unsual color and the extreme sharpness of the blade.

At least three times Tywin Lannister offered to buy Valyrian longswords from impoverished lesser houses, but his offers were firmly rejected. The little lordlings would gladly part with their daughters should a Lannister come asking, but they cherished their old family swords.

In addition to the above, other Valyrian steel weapons known to exist include Heartsbane (House Tarly), Lady Forlorn (House Corbray), Red Rain (House Drumm), and Nightfall (House Harlaw). Brightroar, the Valyrian sword of House Lannister, was lost in an expedition to Valyria centuries ago. An attempt to find it, led by Tywin's younger brother Gerion, apparently ended in failure, with no-one returning from the expedition. House Targaryen also possessed another Valyrian sword, Blackfyre, which was lost in the chaos following the First Blackfyre Rebellion more than a century before the events of the series.

Dawn, the sword of House Dayne, was not a Valyrian weapon but was held to have similar properties (it was forged from a meteorite).

Valyrian steel is vaguely similar to real-life [Wikipedia:Damascus steel|Damascus steel], an exceptionally strong and sharp kind of steel made in ancient and medieval times, but whose method of creation became lost to history in the early modern era. However, Damascus steel isn't actually as strong as current kinds of steel made using modern techniques, but Valyrian steel as it is described in the books is apparently even stronger than modern steel.