Iron pyrite is also known as fool's gold. It is an iron sulphide. It is one of the most common minerals on Earth and is found globally. It is often one of the first minerals that people learn about and is a popular start to a first collection.
Iron pyrite and hematite (iron oxide) are loosely related. In situations where oxygen is plentiful and freely available, iron will form oxides. Iron sulphides will form in anoxic environments. This can be used as a marker signature for the atmospheric conditions on early Earth.
Fossils are often preserved with pyrite, examples include: Beecher's trilobite bed; pyritised billion year old single cell organisms in the Torridonian sandstone in Scotland; and the fantastic pyrite ammonites that you can find in Charmouth and in Grafham Waters. This form of preservation again indicates an anoxic environment. The soft parts of the animal are broken down by sulphur reducing bacteria. The biproducts of this process react with iron in the water to form iron sulphide (again this would not be possible in the presence of large amounts of oxygen -- iron oxide would form instead).
Iron pyrite often makes well formed crystals, these are usually cubes (or octohedrons), or polyhedrons. The polyhedron shapes have a special name they are called pyritohedrons and the faces of the crystals have five sides. Pyritohedrons and cubes have mathematically equivalent symmetries, even though they look very different.