The Rottweiler is named for a town and a trade. In German the dog was the Rottweiler Metzgerhund, the butchers' dog of Rottweil.
Rottweil, on the Neckar in southwest Germany, is very old. The Romans founded it as Arae Flaviae, and Wikipedia calls it the oldest town in Baden-Württemberg. Sources differ by a year on the date. Wikipedia and the newspaper Heidelberg24 give AD 73, and the Reichsstadt Rottweil history site gives 74. Heidelberg24 says the name first appears as Rotuvilla in 771.
The breed's story also starts with the Romans, though that part is tradition more than record. Wikipedia cites the breed standard of the Fédération Cynologique Internationale. It calls the Rottweiler one of the oldest surviving breeds, with ancestors that drove cattle alongside Roman legions crossing the Alps. Britannica is similarly careful: the breed is thought to descend from drover dogs the legions left around Rottweil.
The butchers are better attested. According to Wikipedia, Rottweil's butchers bred their dogs purely for performance and usefulness. The dogs herded livestock, drove cattle and pulled carts of butchered meat to market. That work lasted until the mid-19th century, when railways replaced the old cattle drives.
The town name hides inside the word with a small German addition. Dictionary.com derives Rottweiler from the city's name plus the German suffix -er. English had picked up the word by 1905 to 1910.
Breeders organized soon after. Wikipedia dates the Deutscher Rottweiler-Klub to 13 January 1914 and the Süddeutscher Rottweiler-Klub to 27 April 1915. The two clubs merged into the ADRK in 1921. The American Kennel Club recognized the breed in 1931.
By then the Rottweiler had already helped make another famous German dog. Wikipedia and the American Kennel Club both list it among the likely ancestors of the Doberman, bred in Apolda in the 1890s.
So the name preserves two layers of history at once: a Roman town, and the working dogs of the butchers who lived there.