Born in Gastonia, North Carolina, Floyd received the nickname "Sleepy" playing baseball in the fourth grade, when a spectator yelled "Get that kid out of the game. He’s sleeping." A 6'3" guard, Floyd played competitively at Hunter Huss High School in Gastonia. During Floyd's junior season at Hunter Huss High, he led his team to win in the 1977 North Carolina state basketball championship over rival Ashbrook High, a team led by James Worthy. He also starred at Georgetown University where he is a member of the Georgetown University Athletic Hall of Fame.

At Georgetown, Floyd excelled all four seasons, leading the team in scoring in 1979, 1980, 1981, 1982 and was team Captain in 1981 and 1982. Floyd was named an All-Big East selection in 1980, 1981, 1982, and an All-America in 1981 and 1982. Georgetown's leading scorer in 1979, 1980, 1981, and 1982, Floyd still holds the Georgetown record for Career Points, #1 (2,304) a record that has stood for twenty-three years.

Floyd was drafted by the New Jersey Nets with the 13th pick of the 1982 NBA draft. During the middle of an unspectacular rookie season, Floyd was traded by the Nets with Mickey Johnson to the Golden State Warriors for Micheal Ray Richardson. Floyd quickly blossomed while playing for the Warriors, averaging 16.5 points per game in his first full season with the franchise, and during the 1984–85 NBA season, he averaged a career high 19.5 points per game. Two seasons later, he averaged 18.8 points and 10.3 assists and earned a spot on the 1987 NBA All-Star Team.

In December 1987, Floyd was traded with Joe Barry Carroll to the Houston Rockets for Ralph Sampson and Steve Harris, and Floyd would play 5½ seasons with the Rockets before signing as a free agent with the San Antonio Spurs in 1993, a year before the Rockets would win their 2 consecutive championships. After one season in San Antonio, he returned to the New Jersey Nets, and he retired in 1995 with 12,260 career points and 5,175 career assists.

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