Any limestone or calcite can become cooked or metamorphosed into marble by heat and pressure. Igneous rocks that are intruded by high temperature water can form a lime rich deposit called carbonatite, which is really a derivative of calcite. Sometimes these intrusive formations are nearly 100 per cent calcite.
One common crystal forms of calcite is known as 'Dogtooth Spar.' It appears as a double pyramid. It can be found in cave deposits or can be deposited in gas cavities in volcanic rocks.
Stalactites and stalagmites and most of the tourist attractions in caves are primarily made of calcite. Because it is soluble in water, both the hole which becomes the cave can be eroded, and then the lime can be redeposited in very attractive formations.
There is even an onyx variety of calcite, which is banded with marble like orange, yellow, red, tan, brown or white colours. Don't confuse this with the variety of quartz bearing the same name. Quartz onyx is only black and white. Calcite onyx makes delightful carvings commonly available as bookends, decorative bowls, or even statues.
The calcite variety called Iceland Spar is what most people usually come across first as part of their school science . Iceland Spar not only a beautifully classic cleavage form of the rhombohedron, but it shows the unusual optical property called double refraction. Place a specimen of Iceland Spar over a printed word and when you look through the top you can see the word twice. The unusual optical properties of calcite led to it being used in precision equipment in World War II.
Calcite is made of calcium carbonate. This is a mixture of carbon dioxide, plus water, plus the mineral calcium. It is the common material built up by sea creatures of the extracted carbon dioxide from the water bonded with calcium to make their shells. Most sea creatures make it into a special crystalline form of calcium carbonate, which we call aragonite, which actually lasts longer and is more durable than the ordinary chemically precipitated calcite. That is why fossil shells usually poke out of solid limestone deposits. Under ultra violet light Calcite commonly fluoresces a bright red through to purple or blue, but it is simpler to throw calcite powder into a hot fire which goes instantly a solid orange/red.
A simple chemical test for calcite is the acid test - put anything from lemon juice to vinegar, through to commercial acid on it and it will bubble.
That is why every geologist usually carries a tiny bottle of dilute hydrochloric acid.