(covers information from several alternate timelines)
A hamburger was a variety of sandwich consisting of a cooked beef patty made from ground meat, sandwiched between two buns and accompanied by a variety of other ingredients such as cheese, onions, tomatoes, lettuce, pickles and various other ingredients and condiments depending on taste.
Hamburger with french fries was also an available dish in the mess hall aboard Enterprise NX-01. In August of 2151, a crewman was eating this dish in the mess hall, seated behind Travis Mayweather who had lunch with Matthew Ryan. (ENT: "Fortunate Son")
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 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")