So I think I wasn't quite careful enough with my terminology, so I'll define some terms;
Blocking - pretty much what you think it is, this is when you STOP a blow (be it with a blade or a shield)
Parrying - parrying is bit of a broader term, which some feel includes blocking, but it also includes things like deflecting a blow
Deflecting - pretty self explanatory, redirecting a blow without taking the full force of the attack
Now, most styles have a mix of blocking and deflecting, but to be clear, most 'straight edge' styles make use of both blocking AND deflecting, though the focus is usually on deflecting.
Also, when I said which do you block/parry with, I meant primarily, because yes, there will usually be a bit of a mix.
No matter the style, you will never block with the flat (without otherwise bracing it, against a hand or arm), because it simply does not work, you will be blasted through (though it is very unlikely that the sword will break). If you are stopping a blow, you will always need to use the edge. As others have mentioned, this is always done at the base of the blade, or the 'forte'. Many blades are in fact not sharp here at all.
As for deflecting, deflecting can be done with the flat, but it is both easier and stronger to deflect with the edge, making it a much more viable technique. Primarily (at least in the styles I am familiar with, English, German and Italian styles), edge-to-edge is the only way to go, and that includes systems with an object in the offhand (a buckler, shield or dagger).
There are definitely styles that primarily use the flat though, such as sabre styles and other non-straight edge blade styles (because these blades are typically both shorter and lighter), which is why I specified straight edge swords in the OP.
As for a few things I've been reading along the way:
A few people have said you should be avoiding rather than blocking or deflecting; the only reliable way to avoid a blow is to stay out of distance, and while it is possible to do things such as wait for someone to swing and miss and then come in to attack, it is very difficult to do as your timing must be absolutely perfect, there is no room for error, and it will not work against most competent swordsmen (because they'll see it coming and prepare accordingly). This is because it is very easy to change the direction of a blow when its in motion, by simply rotating the wrist. And despite what some people think, sword play is VERY quick, especially two handed longsword styles (I've watched some longsword bouts where it is almost impossible to track the blows with your eyes). What this means is that when you come in close to attack them (as you must, because your sword is only so long), you need to either make sure you're already giving them an attack to deal with, or lying in a guard that will stop or deflect any incoming blows.
Some others have said that western swords were designed with shields in mind; this is definitely true of earlier swords (particularly viking swords, because they have tiny quillions which fit behind a round shield quite nicely), however later on many swords were either being used by themselves one handed (particularly basket hilted backswords) or being used two-handed (longswords), and so techniques for effective parrying were developed.
A couple of people have mentioned thrusting; thrusting attacks are very powerful, and very easy to turn aside (you can seriously palm aside a thrust). In my experience, one must never use a thrust (especially as an opening attack) unless they really know what they're doing, and even then there are many defenses against thrusts. Some people argue that a thrusting weapon like a rapier, with its superior length, will defeat cut-thrust swords, and this is definitely not the case (if both swordsmen are competent, it's really about even).
Other people don't think you can't block a blow against a stronger opponent, and being one of the smaller and weaker guys in my class, I can promise you this is not the case, a good strong block will still stop them clean.
I should note that at my historical fencing school, all of the techniques we learn come from historical manuals.
Also somebody talked about Katanas; you should never, ever block directly with a Katana, they are not designed for it, you should only defect with the back of the blade (which is reasonably thick).
EDIT: I should also note that I do actually use these techniques in full-contact steel-on-steel bouting, and they do work (they just work better for the people who've been there longer =P)
