Like many other motorcycle manufacturers, Ducati didn't start out that way. Founded in 1926 by the brothers Adriano and Marcello Ducati, it was an Italian radio component production company called "Societa Scientifica Radiobrevetti Ducati." During the 1940s, they made the natural transition to military electronics. After the war, they were aided by the Italian government, eager to rebuild. They tried to go into cameras, producing pocket cameras years before Kodak, but it didn't catch on.

In 1944, a lawyer named Aldo Farinelli had made a 4-stroke 48cc motor designed to mount on a bicycle frame. He named it the "Cucciolo" ("puppy"). In 1946, Ducati put this into production as a clip-on bicycle engine. This was the beginning of the Ducati motorcycle era, followed by:

- 1954: Fabio Taglioni hired
- 1955: Tagioni's 100cc Gran Sport introduced; wins several long-distance events
- 1956: Desmodromic engine introduced; 125cc factory GP racer wins its first race
- 1961: First street-legal Ducati 250 debuts
- 1963: Diana Mark 3 Super Sport introduced in US; fastest 250 street bike in existence

The rest is all recent history, and I leave it for someone else. You can learn more than you want to at http://www.fp1.com/archives/fall/ducati/history/hist.html -- there are also several books about the founders, engineers, and various owners of Ducati, along with its trials and successes.