1. Origins in a medieval culinary practice popular among the Tartars, who shredded the low quality (i.e., tough) beef. Whence we get steak tartar (they didn't use capers and eggs though).

2. Russian Tartars introduced it to Germany prior to the 14th century. The Germans mixed in regional spices and the dish was eaten both cooked and raw and became common among the poorer classes. In Hamburg, the dish acquired the name "Hamburg steak" during a visit by a visiting Irish chieftain who was an ancestor of John F. Kennedy (well, I might be joking about the Chieftain part).

3. The dish went to England and met with Dr. J.H. Salisbury. This guy was a food reformer and physician (popular combo in those days, it seems) who was big on shredding all foods prior to consumption to improve digestibility. He was also big on eating beef 3 times a day, washed down by hot water. Now we get Salisbury steak which was no big deal until Swanson over promoted it (well I might be semi-kidding about Swanson).

4. Concurrently, Hamburg steak traveled with German immigrants to the US in the 1880s and we get "hamburger steak," then "hamburger." The timing of the bun business is not known though Panati says that by the 1904 World's Fair in St. Louis, it was already a sandwich and was also known as "hamburg."