In "Hardhome", the eight episode of season five of Game of Thrones, Jon Snow battles a White Walker. As established, White Walkers can only be killed by dragonglass. But in this battle Jon Snow kills a White Walker with his sword. Does this mean that Jon Snow's sword is made of dragonglass? If not, how does he kill a White Walker?
His sword is from Valyrian steel, which also can kill White Walkers:
Valyrian steel is the only thing that can kill White Walkers, alongside dragonglass, although this property is not widely known, apparently, not even to the White Walkers themselves.
This has been mentioned in the books, but not in the previous episodes of the show:
In "A Feast for Crows", Sam Tarly tells Jon about old annals claiming that dragonsteel (perhaps a synonym of Valyrian steel) is lethal to the Others, like dragonglass. In the show, this is confirmed when Jon Snow kills a White Walker during the battle of Hardhome. In the novels, however, it has not been put to test yet.
In the episode 10 of season 5:
Jon and Sam are discussing the event described in the question and they agree that Jon's sword Longclaw was able to kill the White Walker because it's made from Valyrian steel.
The other answer is a bit too definite; like the characters in the show/books, we know far too little.
The Others/White Walkers haven't been seen for eight thousand years. Most people do not believe they exist. All that we know for sure is what we have seen: normal weapons do not harm them, but dragonglass and Longclaw do.
It is too soon to say whether every Valyrian steel weapon could kill the Others. But fans do love to theorise. Maybe every Valyrian steel weapon would be able to kill them. Or maybe if Jon is the Last Hero or Azor Ahai then it's more about him than the sword. Maybe Longclaw is, or will somehow become, Lightbringer. Maybe the direct fire of dragons will kill them.