Hungarian Paprika is a spice made from the finely ground fruit of what we know as the pepper plant (Capsicum annuum L.). In German, "Paprika" also means bell pepper. Whole, hot peppers are found mostly pickled in Germany and are called "Peperoncino" and "Peperoni" (jalepenos or chilis). Dried peppers are called by their names such as "Thai Chili."
Ground paprika came from South America to Spain in the 15th century. In colonial times, the pepper was introduced to Africa, China, India, Japan and Thailand. The Ottomans took it to Turkey, which borders Hungary in the north (the Balkans). It grows well in the hot, dry plains of Hungary, and the cities of Szeged and Kalosca are famous for their paprika. The first mention of "Paprika" in Germany is found in 1542, where it was used as a container plant.
"Paprika Pulver" or "ungarische Paprika" (Hungarian ground paprika) comes in several different styles. The brightest red is called "Extra" and is mild. Then you find "Delikatess", "Edelsüβ", "Halbsüβ" and "Rosenpaprika" (or "Rosenscharf"), each hotter than the last. Sometimes, very hot paprika is called Cayenne pepper, but true Cayenne pepper is ground from a slightly different pepper plant.
Curiously, paprika does not develop any flavor until it is heated. That is why most recipes start by heating the spice in a little oil, then adding the rest of the ingredients. The bright red color adds an attractive garnish but no flavor to many dishes, such as deviled eggs.
Hungarian paprika is a popular ingredient in German cooking. It is used to make goulash, especially Szegediner Gulasch which is made with sauerkraut, Obazda, a cheese spread popular in Munich beer gardens, and Zigeunersoße, a hot sauce made with bell peppers and ground paprika.
Ajvar is a paste made from Hungarian chili peppers (or chili peppers and eggplant) and is known in German cooking, as well as used as a bread spread or meat accompaniment.
The Guide to Home Cooking has a page on the History of Paprika.