Fluorite is one of the most widely collected crystal minerals, and like quartz, another common element of rock collections, it comes in a variety of colors, some of them quite deep and beautiful. It can even be black or colorless---and its purple variety has been mistaken for amethyst. With so much variation in color, it can be difficult to identify fluorite.
Study the structure of the crystals. The simplest identification mark of fluorite is its cubic crystal structure, which is nearly always present and visible to some degree, even to the unaided eye. This cubic structure is often partially modified by octahedral, or sometimes even dodecahedral or more rare geometries. These modifications will usually show up along the edges of cubes, where an edge will be replaced by an additional face or faces. Octahedral cleavage is an important indicator of fluorite.
Examine the crystal for fluorescing, a hallmark of fluorite. In fact, this quality gave fluorite its name, as it was among the earliest minerals studied displaying fluorescence. Under a UV light, fluorite is most commonly blue, attributed to the presence of europium ions. However, fluorite can fluoresce into other colors as well, such as yellow, green, or red.
Look for "phosphorescence," a glowing after the source of light has been removed, a trait that is rare in fluorite samples, but rarer still in other minerals. Fluorite seems to just like glowing: it can also display thermoluminescence, which is glowing when heated, and triboluminescence, glowing when struck. These latter properties are not found in all fluorite samples, but if your sample does have them, it is extremely likely to be fluorite; few other minerals display as such, and those that do are generally distinguishable from fluorite in other qualities.
Apply a knife to the mineral. Fluorite is a fairly soft stone, only 4 on the Mohs hardness scale. This makes it softer than quartz, and easily scratched---a key reason that it is rarely used in jewelry, despite its great beauty; the other reason is its easy cleavage.
Search for a display of "color zoning," swaths of different colors within the same sample---another distinctive feature of fluorite. While the variety of coloring can be a liability in fluorite identification, it also contributes to one of its distinctions. Large samples almost inevitably display this feature, especially the banding of color.