My favorites are a natural mountain blue stone, MinoSharp 1k, Shapton 2k, Shapton 15k and a natural Tsukuba awase waterstone (pale brown tanba).
For garden tools, axes etc, Norton silicon carbide stones, India stones... cut decently in softer steel and don't wear too much. For curved blades, tool and die maker's finishing stones.
For reprofiling, I like coarse diamond plates or coarse waterstones (but they wear very quickly)
Sharpening harder carbon steel Minosharp 1k followed by mountian blue. For stainless steels the Shapton 1k and 2k
Polishing carbon awase followed by a stropping on a Shapton 15k. Hard stainless, Suehirio 6k or 8k and Shapton 15k, soft stainless Shapton 5k
For carbon damascus and to bring up the quench lines (or hammon), I like natural waterstones (they leave an open pore structure that makes it easy to see the patterns in the steel)... would be a toss-up for stainless damascus though (most natural waterstones and some of the artificial stones will leave a dark dirty looking stain on some types of stainless steels (may or may not be a good thing, depending on the steels and effect you're looking for).
For harder steels, I like softer stones for soft steels I like hard stones. The Shaptons are hard stones, don't form a slurry and are rather hard, but cut quickly... finer stones tend to glaze on stainless but still cut faster than most other stones. The Kitayama 8k is slower and softer than the Shapton pro, but has a nice "natural stone" feel when using it, and the slurry breaks down so it will produce a polish similar to a 12k stone (I haven't tried it, but many recommend it highly) The Naniwa (they are the ones with the pic of the ebi on them... "ebi" is Japanese for prawn/shrimp/lobster) white magnesia stone is also a popular polishing stone. Their colored ceramic stones are popular too but may seem a bit rubbery so probably not the best for the softer steel of kitchen knives (better for polishing up a little blue-steel chip carving knife or something). Too many stones by King, Ice Bear, Ariahiyama, Takenoko, Naniwa, Suehiro, Susin, etc. etc. to try them all, but each type has its own unique properties that may make them really shine for a particular use and not work so well for another use.
There are basically several types of waterstones... there are the magnesia stones that are made by mixing abrasives with a plaster-like binder... they are often white or some light color, are generally soft and can be gouged fairly easily, and may suffer from surface deterioration (often need frequent flattening too), but, while they don't soak up much water they make a good slurry and excel at producing a fine polish and keen edge on hard steel. Suehiro are this type.
Clay or earth baked stones, these are made by mixing abrasives with a clay and baking. They retain water well so are easy to use, don't wear too quickly and are good for general all-round sharpening (can often tell them because the feel and smell like an ordinary red clay flower pot). Many of the King medium and polishing stones are of this kind.
Resin bonded. These use something like epoxy or some sort of resin to bind the abrasives and are baked at a low temperature. They do not soak up water and require frequent wetting, but they can tailor the wear so the binder and abrasives wear at the same rate. These don't produce a slurry and are not as pleasant to use, but cut faster than other types, produce a finish finer than the grit size would indicate (the binder is ever so slightly flexible allowing for a finer scratch pattern) Shapton pro are of this type (Shapton stones do not leave a dark haze on stainless so are recommended for that, they also wear slower than many waterstones but are also more difficult to flatten).
Sintered. These are basically just powdered ceramics like aluminum oxide or silicon carbide, pressed into a brick and partially melted so the grains stick together. They can be very hard and dense like the ruby machinist's stones or very loose like the 120 grit Naniwa green lobster stone. (India and corrundum stones are also sintered, but often used with oil). Crock sticks and Spyderco ceramics are probably of this type (though may use a porcelain type binder as well) The properties of these can be all over the map... some like the 80 or 120 grit Naniwa SiC stones will cut amazingly fast but practically dissintigrate as you use them, while some of those dollar, hardware-store stones hardly seem to wear at all (though clog up and cut slower because they hang onto the abrasive grains long after they've lost their sharp edges).
I don't care for Arkansas stones... while great for things like those good old fashioned Chicago Cutlery carbon steel kitchen knives, they are too soft and slow for some of the hard modern high-tech steels, though the surgical stones make good burnishers (rather than cutters)... there is one kind of black arkansas stone that many people love but is hard to find... it is a mud-stone and should be used with water (old timer's spit on them), rather than the non-porous type that are like using a hunk of frosted glass.
Silicon carbide is harder than aluminum oxide (9.5 mohs) and has sharp crystal edges, but the crystals are fairly fragile and can be broken by tough "grabby" materials like steel. Best for grinding hard materials like glass or weak materials like aluminum or brass. If used on steel, you'll want a stone that wears away as fast as the steel can break off the sharp edges of the crystals off. Comes in all kinds of grits to 1/2 micron but mostly in coarser stones
Aluminum oxide / ruby / sapphire has more blocky hexagonal crystals but are tougher than diamond (softer than SiC at 9.0 mohs but tougher). Not as sharp as SiC but steel will not grab and break the crystals so doesn't wear as fast. Generally leaves a nice even finish. Chromiium oxide is similar to aluminum oxide but a bit softer (about 8.5 mohs). Comes in grits down to 0.05 microns but waterstones usually stop around 1 micron (razor strop powder is about 0.3 micron, and 3M makes a 0.3 micron finishing film as well. HandAmerican's green chrome oxide is 0.5 micron)
Zirconium oxide (zirconia) is mixed with aluminum oxide in grinding wheels and belts (conducts heat well and keep them cooler) It has a hardness between 8 and 8.5 and is used in ceramics (ceramic knives are made of this so diamond and SiC can be used to sharpen them)
Diamond. Hardest (about 4 times harder than SiC) and most wear resistant of all, has a blocky cubic type structure, but the crystals are fragile along cleavage planes so they will break and expose fresh sharp corners. Not recommeded for machine grinding steel because they will dissolve in steel and change the carbon content... cubic boron nitride is only half as hard but does a better job of grinding on steel. Naniwa makes a 6000 grit waterstone but normally the finest diamond plates are about 1,200 mesh or 9 micron. In dust, spray or paste form it can be had as fine as 0.1 micron.
Garnet forms in rounded crystals and can be found in natural Belgian brecca or coticule stones which come between 3000 and 10,000 grit (varies between about 7 and 8 mohs).
Diatomacious earth, novaculite/arkansas stones... a sedimentary rock made from the silica bodies of Devonian diatoms that have been hydrothermally bonded together with a silica cement. The coarser stones often have very pretty bands of color from various impurities. The fine dense stones have fewer impurities (the black ones are colored by manganese dioxide but are still pretty high purity). Hardness is about 5.5 mohs (about that of window glass... steel usually ranges from about 5-pocket knife to 6-a file). The fine mudstone type has a looser bond and will form a mud slurry while the very dense tightly bonded type is almost like a hunk of chert.
Natural Japanese polishing stones have potato-chip shaped abrasive crystals in them (don't remember what they are made of but not as hard as Aluminum oxide). These sort of slice off the microscopic bumps on steel instead of plowing furrows into it like the blocky rounded grains in artificial stones. Since they don't smear the steel around as much, they are less likely to smear over the microscopic pits and leave a more open surface that leaves the structure of the steel more visible. They generally absorb water well and form a fine slurry, and they have a nice feel when using them (you can zone out and hunt for stuff to polish just because the sound and feel is fun). The results are purely cosmetic though and do not justify their high price (about $200 and up) unless you are polishing a sword or a knife and are trying to bring out the texture of the steel and quench lines. Some of them can even come as fine as about 20,000 grit (but such a stone might cost $15,000.00 or more since most of the good ones have been mined out and they are becoming rare).
Everything is a trade-off between cost, the finish they will produce, speed, wear, ease of use (how badly do the stones clog, how often do they need flattening, etc.), and these things vary with what you are sharpening (i.e. an Arkansas stone might be great for that set of Old Hickory kitchen knives, but be useless on D2, ZDP-189 etc. or your ceramic may work great on carbon steel but gum up on stainless).