A hamburger (from Hamburg steak) is a patty of ground beef (sometimes mixed with spices), which is typically grilled, fried, or broiled. It is usually somewhat round, although some restaurants use square patties.

The traditional way of preparing a hamburger (or burger, for short) is to slice a round bun or roll in half lengthwise, then put the cooked patty between the two halves with one's choice of vegetable-based condiments such as:

Hamburgers are also often topped with a slice of cheese, typically Cheddar or "American cheese" (sic), which goes directly onto the hot patty so that it will melt into the surface of the burger. When topped with cheese, it becomes a cheeseburger.

Leave it to a vegetarian to node the obvious. :)