Conrad attended Clifton High School, where he was an All-state quarterback, while leading the team to back-to-back district championships in 1952 and 1953. As a senior, he scored 207 points and took Clifton to the state semi-finals where they lost to the eventual state champion Ranger High School. He accepted a football scholarship from Texas A&M University to play under head coach Bear Bryant. He was a two-way player at halfback and end, although he also played quarterback and fullback. He never was a full-time starter and was a member of the school's 1956 SWC Championship team. Conrad participated in the 1958Chicago College All-Star Game and although he had never attempted a kick in college, he was able to make 4 field goals, 3 conversions, set the game scoring record with 15 points and also intercepted one pass in the 35-19 upset of the 1957 NFL Champion Detroit Lions. In 1976, he was inducted into the Texas A&M Athletic Hall of Fame. In 2002, he was inducted into the Texas Sports Hall of Fame.
In 1958 as a rookie, he was a starter at defensive back and had 4 interceptions and one fumble recovered. On special teams, he made 6 out of 17 field goals and returned 19 punts for 129 yards. In 1959, he was moved to the backfield with college teammate John David Crow. He had 74 carries for 328 yards, 14 receptions for 142 yards, 6 touchdowns, made 6 out of 9 field goals, 18 kickoff returns for 388 yards and 16 punt returns for 133 yards. In 1961, he was moved to flanker pairing him with Sonny Randle, while registering 30 receptions for 499 yards and 2 touchdowns. In 1962, he collected 63 receptions for 954 yards and 4 touchdowns. In 1963, he led the league with 73 receptions and received All-NFL honors. He also had 967 receiving yardsand 10 touchdowns. In 1964, he posted 61 receptions for 780 yards and 6 touchdowns, receiving Pro Bowl honors. In 1965, he had 58 receptions for 909 yards and 5 touchdowns. On June 26, 1969, he was traded to the Dallas Cowboys in exchange for a fifth round draft choice. He left as the seventh All-time pass receiver in the NFL with 418 receptions and had a string of 148 consecutive games played. He came within a game of breaking Don Hutson NFL record for consecutive games with a reception, when he had a string of 93 contests from 1961 to 1968.
Dallas Cowboys
On September 20, 1969, the Dallas Cowboys put injured wide receiver Bob Hayes on the "move list" and activated Conrad. He played in 8 games, recording 4 receptions for 74 yards.
