'Natural' stones:
Usually implies so-called 'Arkansas' stones in the U.S. They are composed of a natural mineral called 'novaculite' (silica-based). Not very hard (Knoop hardness = ~ 825 or so). These only work well with simple carbon steels and low-alloy stainless, like 1095, CV, 420HC, 440A, etc. Elsewhere in the world, other natural stone types exist. Most all natural stones won't be much harder (abrasive-wise) than the Arkansas stones. All are limited to relatively simple steels without much hard carbide content.
Man-made or 'synthetic' stones / other abrasives:
Aluminum oxide (Al2O3) -> these would include Norton's 'India' stones and many other hardware store-type oilstones, as well as many man-made waterstones. Knoop hardness around ~ 2100; suitable for a lot of middle-of-the-road steels with up to moderate wear-resistance (from something like 440C, 154CM, VG-10 up to something like D2, ZDP-189). Doesn't work quite as well with steels containing more than ~ 3% vanadium, the carbides of which are harder than the abrasive (Knoop hardness for vanadium carbide = ~ 2800 or so). Aluminum oxide is heavily used in many polishing applications (pastes, like Simichrome polish, Flitz, Mother's Mag wheel polish, etc). In such form, these also work well as stropping compounds on the same range of steels noted above. (Edited to add: ) Most 'ceramic' hones used in sharpening are also made with aluminum oxide (usually referred to as 'alumina ceramic').
Silicon carbide (SiC) -> Norton's 'Crystolon' is likely the best-known, but also the old-school 'Carborundum' brand as well. Some hardware store stones will also be in SiC (some ACE branded stones, Gator & others), as with the aluminum oxide oilstones. Knoop hardness around ~ 2600; will do better with more wear-resistant steels and alloys with a more vanadium, at coarser-to-medium finish; not as effective on such steels at higher finish/polishing stages, because SiC still isn't quite as hard as vanadium carbide. Most wet-or-dry sandpaper utilizes SiC abrasive, but some wet/dry paper also uses aluminum oxide.
Diamond -> this is the hardest of all, being the hardest reference material on the Knoop scale, assigned a hardness value of 7000. Works THE BEST with uber-wear-resistant steels having a lot of vanadium content. Think of steels like S30V, S90V, S110V, etc. for sharpening on diamond.
Cubic boron nitride (CBN) -> Also works well with high-vanadium steels. Second place only to diamond in Knoop hardness, at around ~ 4500 or so. CBN is more expensive to produce than synthetic diamond though, so stones made from it can be pricey.
Chrome oxide ('chromium oxide', Cr2O3) -> this is the same stuff originally developed as a powdered pigment for green paint. It's what 'green compound' is ordinarily made of, though some 'green' compounds may be blended with aluminum oxide as well. It's less-hard than aluminum oxide (something like Knoop ~ 1700 or so). Ordinarily only used as a metal buffing/polishing compound; works well on simpler steels with not much wear-resistance (1095, CV, low-alloy stainless like 420HC, etc).