Basically a summary of an article that appeared in issue 125 (July/August 1997) of Fine Woodworking. Note that alumina zirconia falls under "ceramic."
Aluminum Oxide
-Best all-purpose paper.
-Cheaper grades are dark brown; better grades are almost white. Some brands are better than others.
-Standard ‘production’ paper.
-Lasts longer than other abrasives.
-Friable, which means that the grains break in use and so constantly renew their edge.
-Typically “open coat” which means that there is space between the grains for the sanding dust.
Stearate Coated
-Typically made with Aluminum Oxide or sometimes Silicon Carbide.
-Coating is typically Zinc Stearate and acts like a soap, cutting down on clogging.
-Really only helps when sanding oily or resinous woods, or finishes.
-Stearate can interfere with water-based finishes.
-Otherwise properties derive from the abrasive used on the paper.
Silicon Carbide
-Harder and sharper than aluminum oxide, which means that it is especially good for hard materials such as paint, plastic, and metal.
-Usually a waterproof, “closed-coat” paper.
-Dulls more quickly than aluminum oxide when sanding wood because wood is not hard enough to break the grains. Also clogs easily if used dry.
Ceramic
-More costly and less common than other abrasives.
-Very aggressive and tough.
-Mostly used for coarse work like shaping.
-As with silicon carbide, wood is not hard enough to break the grains but this is less of an issue with ceramic grit because of its extreme toughness.
-Trade names include Cubicut, Regalite, Alumina Zirconia, Norzon, and AZ.
Garnet
-Light tan colored natural mineral.
-Non-friable and not very tough so it breaks down quicker.
-Produces a softer cut, which makes for a nicer final finish on wood.
-Excellent for final sanding and for end-grain.
-Garnets tendency to burnish the wood can make stain penetrate more evenly, but less deeply on blotch-prone woods.
Other Notes
-FEPA (P-Scale) abrasives are graded more tightly then CAMI (US Standard) graded papers, so they produce a more consistent scratch pattern. Micron grading is even tighter than FEPA. The scales also do not match up exactly, but they are not too far off in the grit ranges typically used for woodworking.
-Paper backings are graded in weight using a letter scale: A, C, D, E, and F (lightest to heaviest).
-Cloth is stiffer but not as flat-backed as paper. It is tougher than paper but produces courser results. It comes in two grades: X (heavy), and J (light).