Under Armour was a fallback name, and it came from a family joke. Kevin Plank's first choice was "Heart," as in wearing your heart on your sleeve. The trademark application was denied. He then tried "Body Armor," and that failed too.
His brother supplied the answer, perhaps without meaning to. In an interview reported by The42, Plank recalled him asking: "How's that company you're working on, uhh … Under Armor?" Plank was not sure if his brother was joking, but the name stuck. Wikipedia tells the same story and does not name which brother it was.
The spelling came from a phone number. Plank chose the British "Armour" because a toll-free vanity number was still available in that form. He said he "thought the phone number 888-4ARMOUR was much more compelling than 888-44ARMOR." He told The42 he was skeptical at the time "whether this whole internet thing would stick."
The product came from sweat. Plank was a special-teams captain on the University of Maryland football team. He was tired of cotton T-shirts that soaked through under his pads, so he looked for a fabric that would pull moisture away from the skin.
Wikipedia dates the founding to September 25, 1996, when Plank was 24. He ran the business from his grandmother's house in Georgetown, Washington, D.C. The42 places it at the corner of 35th and O Street. He lived upstairs, sold from the ground floor and kept stock in the basement.
The first big break was an equipment manager at Georgia Tech, who ordered ten shirts. Sales in that first year came to $17,000.
The word itself is old. Etymonline dates armor to about 1300, meaning protective covering worn in combat. It came from Old French armeure and Latin armatura, "arms, equipment," from arma, "weapons." Plank's twist was to put the armour underneath, next to the skin, and sell it as a T-shirt.