From the Twilight novel, the only way to be sure is to rip the vampire apart, and then burn the pieces. The vampires seem to have known this about themselves since time immemorial. The Quileute werewolves discovered it independently, when one of the wolves brought back pieces of a vampire who had been killing Makah women in a neighboring village. The pieces were still moving and tried to reassemble themselves, until the Quileutes scattered and burned them.

Other than that, there is no known way to really kill a vampire in the Twilight universe. Their skin is hard enough that only two things are known to pierce it; vampire teeth and werewolf teeth. So, no wooden stakes through the heart, no silver bullets, etc etc. They don't ever have to breathe, so drowning, strangling, etc don't work, and crosses, holy water, even sunlight don't affect them in the slightest. So, at least in the books, only a werewolf or a vampire could kill a vampire.

However, Aro, in the last book Breaking Dawn, says that human technology has progressed to the point where humans can actually pose a danger to vampires and even kill them. I can only assume this is a reference to explosives, whether conventional or nuclear. Shoot a vampire with a rocket launcher and you'd be approaching the kind of force a vampire themself would be able to exert. Dropping a thermonuke onto Volterra would probably mean no more Volturi. Obviously speed would still be an issue, but vampires, though fast, seem to still be subsonic, so a guided missile could probably track and intercept them, and if you managed to catch a vampire unaware you have a chance at detonating a weapon like this before they could get out of its range.