Stone has been carved for a long time without much equipment being necessary. Softer stones are, of course, easier to work with. At its simplest, it can be done with a lump of soapstone (steatite, or talc) small enough to hold in the hand, and a shortbladed knife. It comes in lots of colors, and is the softest stone that will hold together well enough to be worth carving. Pipestone, or Catlinite, is another easily worked stone that lends itself to carving with steel tools. Native Americans used it to make their famous calumet "peace" pipes. Steel files will work on these materials for smoothing out, as will any sandpaper. One can also use rotary files, burs, and mounted stones if you have any of these- but they aren't necessary. Roughing out the shape with a saw will save time, but one can also just start whittling. Although these soft stones won't take a very high polish, they can be made shiny by progressing from 120 to 600 grit sandpaper and then using pastewax.
There are some safety issues involved which are important to consider. First, be careful not to cut yourself. Restaurant supply outlets carry butcher's gloves that will protect somewhat against a slicing wound, which are common among those who carve using the hand as a vise. They aren't as effective against punctures, however. Soapstone can contain asbestos, which can cause lung cancer as well as other toxic effects. California soapstone is notorious in this regard. While one can take samples to a testing lab to make sure, this would probably only be worthwhile if one was quarrying one's own. Most stone dealers are aware of this problem, and only carry stone that is known to be asbestos-free. Some soapstone contains silica, which can cause silicosis when inhaled. Silica is a major component of all quartz-based minerals, like amethyst, citrine, flint, chrysoprase, and even granite. It can also contaminate other sorts of stone, like marble. This is definitely something to avoid breathing, but is not considered as serious a hazard as asbestos. It is important to note, though, that freshly ground silica is much more hazardous to inhale than dust which has been allowed to hydrate in the atmosphere. Freshly fractured silica carries a surface charge that makes the particles much more membranolytic or cytotoxic. If they are allowed to hydrate in the lungs, silicic acid will form, which can be acutely toxic. Sandblasting, for instance, which liberates large amounts of crystaline free silica, has been known to kill people doing it after a short period of intense exposure.
If one is interested in carving stones containing silica, such as granite, flint, quartz, etc., make sure proper precautions are followed. Wear a dust mask when carving, and clean up thoroughly afterwards with the mask on. Note that power tools, especially abrasive ones, make quite a bit more dust than hand tools, and protecting oneself becomes even more crucial. Safer stones to carve are alabaster (a massive form of gypsum, or calcium sulphate), limestone, marble, and Mexican onyx (which are all basically calcite, or calcium carbonate), although these may also be contaminated with small amounts of silica. These are quite a bit softer than the silica-based stones, and are consequently easier to carve.
Larger pieces of stone, mostly marble and limestone (although granite and even harder stones are still sometimes carved this way as well) are traditionally carved using a succession of steel points and chisels. These tools are struck with a hammer, which can cause metal chips to fly as well as stone: be sure to wear adequate eye protection. The points are used first for roughing out, then toothed chisels are used to remove the faceting left by the points, and to determine the compound curves. Last of all, straight chisels in a range of sizes are used to define details. To begin with, it is best to get some stones of about 2 cubic feet, which is big enough to stay still when one hits it, but not so big that two people can't lift it. (For larger stones, one starts to need some mechanical lifting equipment.)
Fine details can be added with progressively smaller chisels, or abrasives can be used to create a smooth or a polished surface. For the hard stones, chisels with tungsten carbide tips are preferred, as they keep their edges longer than the plain steel kind. These edges are, of course, ground to a more obtuse angle than the edges of woodcarving chisels, and they are much heavier as well. Nowadays sculptors often use reciprocating pneumatic tools to hold the chisels, and electric or pneumatic grinders instead of patient hand-rubbing, but the process is much the same. For stones the hardness of marble or softer, tungsten carbide burs work fairly well too. One type of tool, the "rotary chisel" has recently been invented that has a triangular piece of tungsten carbide mounted on a shaft transversely, with each apex sharpened to a cutting edge. These will not load up with waste like parallel-fluted burs, and they remove material quickly. They are also, however, more dangerous, so one must make sure always to approach the material at full RPMs and to keep ones other hand out of the way.
Although it is slower than using chisels or power tools, carving can be also be done with hand abrasive techniques. This goes quicker with soft materials, but hard stones can be carved this way if one has a lot of patience. Amber, soapstone, alabaster, and the calcites can also be worked with steel riffler files- the ones with odd-shaped ends- then smoothed out with successive grits of silicon carbide sandpaper. For harder stones, one can get diamond files and rifflers to do the roughing out. If one is primitively inclined, a bow-drill can be handy for making holes. These were used originally as fire-starters, but were soon equipped with stone points and became the first "power" tools. Mounted horizontally and powered by foot, they evolved into the first lapidary arbors. With a Jacobs chuck and a drip-feed to attached one of these, one can use it the same way as a flex-shaft tool, with the added advantage of being able to hold the work with both hands. Of course, an electric motor-powered lapidary arbor can also be mounted with a chuck for the same effect.
If one is faced with carving a larger, harder stone, I find that a right-angle grinder, preferably with a water-feed running through the central shaft, has a lot to be said for it. Whenever using electricity and water together, make sure to run the power through a ground-fault interrupter, which cuts off the power when it senses a short- otherwise one runs the risk of electrocution, First, one attacks the stone with a diamond sawblade, making parallel cuts and breaking out the waste. This removes a lot of material quickly. After that, one can use a diamond cup wheel mounted on the right-angle grinder for rough grinding, removing the stubs left from the first operation. Then there are velcro-backed diamond sanding pads that will take large simple and compound curves all the way to polish. Silicon carbide sanding disks can also be used for most of this, and they are considerably cheaper, although they won't last nearly as long, and work much slower on the harder stones.
Steel bits with diamond grit plated onto their surfaces have come down considerably in price of late, and can help get into small places. To polish in these complicated areas, maple or phenolic plastic rods are turned to appropriate shapes, then charged with diamond compound and some grease (olive oil paste is best, but petroleum jelly works. A few drops of mineral oil help keep heat from building up, which can damage delicate stones. Dremel tools have a very limited usefulness, and can't be used wet. Flexible shaft (Foredom, Pfingst, etc.) tools are better, but are still mostly used for very small (fist-size and smaller) work. They are excellent for gemstone carving, but larger tools, such as die grinders, are best for larger projects. They generally hold bits, burs, and stones mounted in a 1/4" (or smaller) arbor, and can be slowed down, if desired, using an inline speed controller.
There are various proprietary flap-wheel type sanding devices made which can be useful for removing the scratches put in by the coarser tools, or one can jam a piece of sandpaper or abrasive cloth into a split shaft- a cylindrical piece of steel split lengthwise for about half its length- to get into tight spaces without danger of digging in or overheating. Marble is often polished using an abrasive mixed with an acid to speed up the polishing action- be careful of this; although it works, it can cause burns and other nasty health effects. There are also many different polishing compounds, besides diamond, which are used to polish stone, usually these are mixed into a slurry or paste with water, then applied to a hard felt wheel or point tool. These must generally be recharged periodically to avoid overheating. Of course, polishing is difficult and time-consuming; one must decide if it is necessary for the piece in question. Roughly carved stone is often acceptable as part or all of a piece, although ornamental stones won't show their colors and patterns to full advantage without polishing