Moore was born in Tallahassee, Florida in 1951. He grew up in Miami, Florida and attended Miami Edison Senior High School and Miami-Dade Community College.
On the recommendation of his junior college football coach, Moore received an athletic scholarship to transfer from the University of Tennessee at Martin to the University of Florida in Gainesville, Florida, where he was a star running back for coach Doug Dickey's Florida Gators football team in 1972 and 1973. As a junior in 1972, Moore led the Gators with 145 rushes for 845 yards and nine rushing touchdowns, twenty-five receptions for 351 receiving yards and four touchdown catches, and 230 return yards, while earning first-team All-Southeastern Conference (SEC) and honorable mention All-American accolades.
Moore graduated from Florida with a bachelor's degree in exercise and sport science in 1975, and he was inducted into the University of Florida Athletic Hall of Fame as a "Gator Great" in 1978. In a 2006 article series published by The Gainesville Sun, the newspaper's sportwriters ranked him as No. 49 among the 100 all-time greatest Florida Gators of the team's first 100 seasons.
Moore was chosen by the Miami Dolphins in the third round (seventy-eighth pick overall) of the 1974 NFL Draft, and he played for the Dolphins for thirteen seasons from 1974 to 1986. He was elected to the Pro Bowl in 1977, after a season in which he made fifty-two receptions and led the league with twelve receiving touchdowns (he also had a rushing touchdown that year). Moore is immortalized in the famous "Helicopter Catch" video clip—while making a reception against the New York Jets in Giants Stadium in 1984, he was hit simultaneously from opposite directions by two Jets tacklers sending his body spinning into the air. The catch was a crucial third-down conversion, leading to a score and a come-from-behind win in a closely contested divisional game.
By the time Moore retired at the end of 1986, his thirteenth season with the Dolphins, he had broken almost every receiving record of the Dolphins; his team records, however, were subsequently broken by Dolphins wide receivers Mark Clayton and Mark Duper in the 1980s and 1990s.
His final career receiving statistics were 510 catches for 7,547 yards and seventy-four touchdowns. He also rushed for 249 yards and a touchdown, returned twenty-seven punts for 297 yards, and gained 856 yards on thirty-three kickoff returns.
Moore is also known for his humanitarian work and philanthropy. In 1984, the NFL voted Moore as "Man of the Year," an honor given to a player who gives outstanding service to his community. Moore also received the Byron White Humanitarian Award in 1986. He created The Nat Moore Foundation, an organization through which he continues to work with disadvantaged youths in the Miami-Dade County area, in 1998.
On December 5, 1999 he was added to the Miami Dolphins Honor Roll.
Moore was a football broadcaster for Florida Gators football games on Sun Sports until 2011. As an announcer, he was notorious for adding an "s" to the last names of various players (Chris Leak became "Chris Leaks," Percy Harvin became "Harvins," etc.) In addition, he teams with Bob Griese to provide television analysis of preseason Dolphins games. He also owns a sports promotions firm, Nat Moore & Associates, Inc. He is a vice president in the Miami Dolphins organization and oversees the Miami Dolphins Alumni Association, and also serves as the executive director of the NFL Super Bowl Football Clinic.