Calcite is a very common mineral in many different geologic settings; it makes up most limestone and marble, and it forms most cavestone formations like stalactites. Often calcite is the gangue mineral, or worthless part, of ore rocks. But clear pieces like this "Iceland spar" specimen are less common. Iceland spar is named after classic occurrences in Iceland, where fine calcite specimens can be found as big as your head.
This is not a true crystal, but a cleavage fragment. Calcite is said to have rhombohedral cleavage, because each of its faces is a rhombus, or warped rectangle in which none of the corners are square. When it forms true crystals, calcite takes platy or spiky shapes that give it the common name "dogtooth spar."
If you look through a piece of calcite, objects behind the specimen are offset and doubled. The offset is due to refraction of the light traveling through the crystal, just as a stick appears to bend when you stick it partway into water. The doubling is due to the fact that light is refracted differently in different directions within the crystal. Calcite is the classic example of double refraction, but it's not that rare in other minerals.
Very often calcite is fluorescent under a black light.