The McDonald's breakfast sandwich consisting of a sausage patty between an English muffin is the Sausage McMuffin.
A standard Sausage McMuffin contains a seasoned sausage patty served on a toasted English muffin. The basic version does not necessarily include an egg; adding an egg creates the separately named Sausage McMuffin with Egg. That distinction is why Egg McMuffin is not the best answer to this question.
The McMuffin family grew out of the Egg McMuffin, developed by McDonald's franchise owner Herb Peterson in California. Peterson introduced the original sandwich in 1972, using an English muffin as the bread and drawing inspiration from eggs Benedict. Sausage and chicken versions were later added to the McMuffin line.
The sandwich is also easy to confuse with a Sausage Biscuit, but the bread is the key difference: a Sausage McMuffin uses a round, griddled English muffin, while a Sausage Biscuit uses a soft American-style biscuit. McDonald's breakfast menus can vary by country, but the name “Sausage McMuffin” is the canonical U.S. menu term.