A hamburger was a variety of sandwich consisting of a cooked meat patty made from ground beef, sandwiched between two halves of a bun, and accompanied by a variety of other ingredients such as cheese, onions, tomatoes, lettuce, pickles, bacon, and various other ingredients and condiments depending on taste. It is usually served with French fries and Soda pop.
A hamburger with cheese was called a cheeseburger. There were a variety of burgers, one of which was a chili burger, which was a hamburger covered with chili. (ENT: "Carbon Creek")
In an alternate version of 1944, Carmine offered Alicia Travers a portion of hamburger meat near the Sixth Avenue subway station. (ENT: "Storm Front")
In 2004, Detroit's Burgerland prepared various styles of hamburger, including the Double Superbeef and Triple Superbeef. (ENT: "Carpenter Street")
Hamburger with french fries was also an available dish in the mess hall aboard Enterprise NX-01. In August 2151, a crewman was eating this dish in the mess hall, seated behind Travis Mayweather who had lunch with Matthew Ryan. (ENT: "Fortunate Son")
A hamburger with a side of French fries was among multiple dishes that, in 2153, Tarquin served Human Starfleet Ensign Hoshi Sato, while she was on his planet. (ENT: "Exile")
In an alternate reality in 2255, Shipyard Bar in Riverside, Iowa, on Earth, offered Earth burger as a food item on its menu. (Star Trek)
In the late 2250s, a cheeseburger was one of the many food items vomited out by food synthesizers when Me Hani Ika Hali Ka Po and Sylvia Tilly were in the mess hall aboard the USS Discovery. (ST: "Runaway")
In 2257, when Una Chin-Riley beamed aboard the USS Discovery from the Enterprise, she ordered a cheeseburger with habanero sauce and fries on the side. (DIS: "An Obol for Charon")
In 2371, Neelix concocted his own style of open-face cheeseburger. Tom Paris suggested that he should give it an original name, suggesting the "Double Talaxian with Cheese" or the "Greasy Neelix", which Neelix said he would take under consideration. (VOY: "Fury")
In 2384, the former crew of the USS Protostar ate cheese burgers in the mess hall of the USS Voyager from time to time, especially double cheese burgers. (PRO: "Observer's Paradox", "Ascension, Part I")
Kathryn Janeway often ate cheeseburgers in the mess hall of her ship as well. She once ate a burger at a table with Chakotay. (PRO: "Ascension, Part I")
According to the Star Trek Encyclopedia, 4th ed., vol. 1, p. 135, the cheeseburger was often associated with the American culture.