The time is now,
Global Thermo-Nuclear Warfare.
Unless the target is a cockroach.
Do the job in front of you.
To Aprilla, I think it my friend really meant it as breaking the trachea with a knife-hand strike in my opinion, karate chop in the throat just sounds funnier and it's close enough. She also mentionned kicks in the balls, but noted that it's only lethal if you hit very hard.
Assuming "fastest", you're looking at either throat-slitting with a knife (since that makes the person too busy choking to death to bother fighting you) or anything involving head wounds. If you have a hammer, pounding someone's head to pulp is difficult, but if you hit them hard enough the first time they pretty much stop, and so you can finish them off. (Note that this requires a good solid blow or a big hammer, preferably both; a sledge will stop someone dead, but a hand-held hammer might not work quite so well).
Or, you can slam a knife through their eye. That'll take them out pretty much instantly if the knife's long enough and actually goes through to their brain. But that's a bit... unlikely to happen.
Bullet to the head; simple, quick, and efficient.
^ People have survived cannon fire to the head before.
No really, one guy in the ANA took a 20mm round to the brainpan and lived.
edited 27th Mar '12 5:53:03 PM by MajorTom
Lucky for us, those people are part of a very small, and very lucky minority.
Yeah, generally you don't need to worry about those guys...
But the guy did say "without a gun"; I'd assume cannons count. If they don't, of course... ...
Shadowed Philosopher
Technically, 'gun' refers only to a specific type of artillery that attempts to destroy its target with direct fire by the momentum of the round; by that definition, the most efficient way to kill someone is "have a pistol; target center of mass; fire multiple times". I'm assuming they meant the more general vernacular usage of 'firearm'. 
Wait, query. Most efficient way... hmm. Wouldn't that be a headshot? The center of mass is certainly the easiest way to kill someone with high efficency, but it is not the optimal route. It is the optimal route that most people could do, but that doesn't mean it's completely optimal.
Though I agree that on pure "actually doing it" terms, that one wins. Someone with higher capabilities has other choices.
ETERNAL
As Aqueos said, a simple beheading would do the trick, If you want the beheading to be Efficient however, You'd probably want to go the way of the Guillotine if you have access to one, since with the standard choppy choppy with an axe on a stump, or the choppy choppy you're knelt down and I chop with my sword at your neck, sometimes the head doesn't come off completely in one go. Of course you could just hang them, if you have preparation, good mathematical skills, a good hanging would break their neck, a fast and semi-efficient kill. Semi because, well, you need to prepare a gallows and such.
"You'd never do something as irrational as dying."
Book Fort is Best Fort
When I think of something that can kill quickly, the first thing that comes to mind is deadly neurotoxin. Preferably the kind GLaDOS used.
She preferred to use a gaseous neurotoxin that could kill a man quickly and painfully.
But really, I'm sure that a deadly neurotoxin or some other poison could kill quickly if it was injected directly into a sensitive spot, like the spine, or inhaled in large amounts in the case of poison gas.
Okay, fine! I'm going to do something I'm sort of good at!
Happy New Year!
Or just running them over with a car. Remember K.I.S.S. people.
hashtagsarestupid
Stop Killing My Titles
If we're taking quality over quantity, there's no need to rush. Apply some poison, watch from the safe distance as your victim coughs up his or her lungs.
If we want something flashy, yet extremely effective, The Ark Of The Covenant is a perfect tool.
If we want to be absolutely sure that our victim will die, separate their head from rest of their body, eventually remove their heart. Unless they have some super regeneration powers or just No-Sell such a trick, it should do fine.
grah
Snapping a neck so it contorts the brain stem. About as clean and practical as you cn get. If your character doesn't have Super Strength, then an instant kill poison or an icepick to the neck is also fine.
edited 11th Apr '12 12:26:40 AM by gameknight102xx
Well if you want them dead and practicality is no huge object, just go for a coup de grace beheading.
Impractical, but then they are 100% dead instantly.
Bet you didn't see that coming