Picture jasper: when the stone looks like a painting
This is a favourite for many people, and for good reason. Picture jasper is characterised by its bands and patches in brown, beige, ochre and sometimes greenish tones, which resemble hills, horizons or desert landscapes seen from a distance.
This "landscape" effect happens because, during its formation, layers of sediment with different concentrations of minerals (mainly iron and manganese oxides) were deposited and later "frozen" into the stone like miniature geological strata. In a sense, each piece of picture jasper is literally a map of how that sediment was deposited, layer by layer, as it formed.
Leopard jasper: spots that aren't random
Leopard jasper owes its name, as you can probably guess, to its rounded dark spots on a lighter background, resembling a leopard's coat. These spots usually form from localised concentrations of minerals (often manganese or iron oxides) that clustered into "cores" during the rock's formation, rather than spreading out evenly.
It's a great example of how the same type of "ingredients" (metal oxides) can produce completely different visual results depending on how they're distributed: evenly (red jasper), in layers (picture jasper), or in concentrated spots (leopard jasper).
Dalmatian jasper: straight out of a Disney film
Dalmatian jasper is perhaps the easiest to identify at a glance: a white or cream background with rounded black spots, exactly like a Dalmatian's coat. There's an interesting nuance here: although it's commonly sold as "jasper", from a strict mineralogical standpoint Dalmatian jasper usually has a somewhat different composition from "classic" silica jaspers, often consisting of a feldspar base with black tourmaline inclusions (those famous spots).