A stalactite ("c" in the middle) is a growth of crystalline calcite precipitated from solution in the ground-water feeding it, and growing downwards from a water-carrying joint in the roof of a limestone cave.
Its opposite number, growing up from the floor, is a stalagmite.
Stalactites hang from the ceiling of caves and are formed by mineral-rich water dripping down, while stalagmites grow up from the floor of caves from the mineral deposits left behind by dripping water. Stalactites have a tapering icicle-like shape, while stalagmites are more mound-like and grow upward.
They typically form in caves where they can grow undisturbed for hundreds of years. Stalagmites grow from the floor up and stalactites stick to the ceiling. Dripping water deposits minute amounts of limestone that build up over the years. +++ You do sometimes find small, rather coarse versions growing under bridges etc.built from bricks or ashlar masonry and mortar. Here, the calcite that is their constituent mineral has been leached from the structure's mortar. However caves in limestone (the rock hosting by far the majority of caves) are the natural home for stalagmites and stalactites, and their related formations.