Since there's been such a good response to this idea, here's the first post. I'll only be trying to make one point in each one, to keep things clear (and to give me plenty of posts while I'm caught up with other things). Point one is simple: the sword's construction matters.
To put it another way, a katana is not used in quite the same way as a european longsword, or a rapier, or a Chinese Jian. It is not designed to be. It is used primarily two handed (though I'll be posting on that shortly) with pushing or drawing cuts in a slicing action, rather than the neat thrusts of a rapier or the hacking strokes of a machete. It is much sharper than something like a longsword (where grabbing the blade is a workable possibility with a gauntleted hand), but also much lighter and less likely to get through steel plate.
You might think it can't make that much difference, and that many things would be the same. That is true at a basic level, and we will discuss them too in due course. Yet the differences matter. Take me when I'm fencing. I fence sabre. I'm good (well, not that good. 247th in the british rankings at the last count) with a sabre. Yet I am far less effective with an epee (it's the stabbing and the emphasis on bladework) and positively rubbish with a foil.
Far too often, though, people write sword scenes as though it doesn't matter what the hero is using. They pick up a completely unfamiliar sword and they know how to use it. They do the same slashes, thrusts and parries, regardless of whether they're using a short Roman gladius or five feet of German two handed sword. So the next time you're writing, think about the weapon your character is using. Are you getting its feel over? Is it the right feel for the character?