In Simple Terms
he term edge geometry, when applied to cutting sword blades means the shape of the cutting edge. If you want to buy swords, this is important because the edge geometry defines how the sword will cut. This is in combination with the weight, balance, polish and resilience of the sword. A sword won't necessarily cut well just because the edge is sharp. In fact, even good Chinese swords may not cut well if the swordsman approaches the cut with a poor edge angle. There is a very interesting discussion of how the edges on different swords respond here, on the GRTC forum.
This is the thinnest and sharpest edge. It blunts quickly and is easy to sharpen. It can also roll easily because there isn't much weight of steel behind the cutting edge.
This edge is flat on one side and angled on the other, so it is like sharpening only one side of the edge.
This is probably the most common and is frequently used on knives. It angles evenly from both sides to a sharp point. This is a very sharp edge for a sword but does not dull or roll as easily as a hollow grind.
Appleseed Grind (also known as convex, bullet, or sabre grind)
This is the edge almost always used on ancient Chinese swords. It curves gently to the edge and has the most weight of steel behind it. It is a very efficient cutting edge as long as the swordsman has good control of his angle of entry and exit to the cutting target. Since less of the edge is honed thin. it is very unlikely to roll or dull quickly. On the other hand, more care is needed when sharpening the sword if the bullet shape is to be maintained.
Those are the basics of blade geometry. There are also bevels - planes on the way to the edge. People who specialise in sharpening blades know which bevel, secondary bevel or micro bevel and which degree of angle is correct for the purpose of each cutting tool. They use specialised equipment and even have precise methods of ensuring they have flat surfaces. For extra information on blade geometry, I suggest KnifeArt.com
If you wish to sharpen your own sword, you'll most likely end up with flat geometry unless you have some specialised knowledge. It's best to know what sort of edge geometry your sword started with before deciding to attempt this. The new Hanwei Cutting Jian designed by Scott Rodell, uses the appleseed edge to keep the design authentic.