People use stone to build walls, buildings, roads, bridges, wells, and many other things as well. While it's possible to build walls and other structures using stones found on the ground, in whatever shape they happen to have naturally - see the article on dry stone walling - most of the time people who build with stone want the stone cut into a particular size and shape - most often into rectangular blocks.
The basic tools for cutting stone are the pick, sledge and chisel. They are generally made from the hardest material available, often iron or steel, but in places where iron is unavailable people have used bronze or stone tools. Tools used to cut stone tend to quickly become blunt and need frequent sharpening or re-forging.
A pick is a bit less than 1 meter long, with a handle usually made of wood and a head usually made of iron or steel. The head has two sides; most often, one side narrows to a point, while the other is flat, like a garden hoe or a scraper. At a quarry a pick is used to clear away loose rock and other debris. It can also be used to prise apart a stone that already has a fracture.
A typical sledge is about a foot long, with a wooden handle and a head made or iron or steel. It is used for striking a chisel, and can be used to deliver either careful little taps, or powerful blows.
Terminology note: a mallet is similar to a sledge, but is used to work with wood, not stone. A sledge is sometimes called a hammer, but a hammer usually has a flat, sharp edge that can be used for prying, while a sledge does not. A larger version of a sledge, with a longer handle, is called a sledgehammer; it is used to deliver powerful blows to smash things apart.
A chisel is about a foot long, and usually made of iron or steel. One end is designed to be hit with a sledge, while the other end is designed to cut into stone. Different kinds of chisel have different cutting ends.
The first step in splitting a stone is to drill holes in it. Before power tools became available, this was often done using a star chisel; a stone-cutting chisel with a ring of triangular 'teeth'. To drill a hole into a stone, the stonemason would place the chisel against the stone, strike the chisel with a sledge, then rotates the chisel 1/8 of a turn or so, and strike it again, repeating over and over. This must have been exhausting and time-consuming work.
The basic process to split a stone is: first, draw a line on the stone where you want it to split. Second, drill a series of holes into the stone along the line. Third, drive a wedge or chisel into each hole and tap it lightly with a sledge. Repeat this over and over until, finally, the stone splits along the line.
A variation on this is the plug and feather method (also called 'plugs and wedges' or 'feather and wedges' or 'wedges and shims' or 'pins and feathers'). A line is drawn across the stone and holes are drilled along the line, and a plug and feathers is inserted into each hole. The feathers are steel wedges that get wider at the bottom, while the plug is a steel wedge that narrows at the bottom; the plug is inserted between the two feathers. Each plug is gently tapped with a sledge. This is repeated until the stone finally cracks and splits along the line.
It's hard to imagine a stone being split in half. Videos can help:
The types of stone most often used in building are granite, which is very hard, and limestone, which is softer. Marble is a type of limestone. Slate repels water particularly well, and is used to make roofing tiles. Sandstone is very soft, and is used for carving sculptures.