Paprika is a spice that comes from a type of sweet red pepper sometimes known as the paprika pepper (Capsicum annum). The seeds are removed from the fruit and the peppers are then dried and subsequently ground down into a deep red powder. Originally from Spain and the West Indies.

There is Pimentón from Spain. As Kristen pointed out, chile peppers are grown, dried (sometimes over a wood fire to impart a smoky essence) and ground. There are basically three types in Spain, hot, sweet and medium (agridulce). There is also Hungarian paprika, which is a milder form.

Paprika comes from grinding dried peppers -- either sweet or hot. (Some varieties are smoked in the meantime too -- extra delicious!)