"fist-fighter, pugilist," late 15c., agent noun from box (v.2). The breed of dog (1934), is from German Boxer (the breed originated in Germany), itself taken from English boxer "fighter;" the dog so called for its pugnaciousness. Boxer shorts (1943) so called from their resemblance to the attire worn in the ring.
"fighting with the fists as a sport," 1711, verbal noun from box (v.2). Boxing glove "padded glove used in sparring" is from 1805.