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.
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 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")