Morris played minor hockey in the Alberta Midget Hockey League with the Red Deer Chiefs and was awarded the Brian Benning Trophy as the league's top defenceman in 1995. He went on to play major junior with the Regina Pats of the Western Hockey League. After his rookie WHL season, in which he scored 52 points in 67 games, Morris was drafted 13th overall by the Calgary Flames in the 1996 NHL Entry Draft, a somewhat surprising pick as Morris had been projected to be a third- or fourth-round pick. He returned to play a second season with the Pats, improving to 18 goals and 75 points before turning pro. Following Morris's 1996–97 WHL season with the Pats, he was assigned to the Calgary's American Hockey League affiliate, the Saint John Flames for the final seven games of the regular season, as well as five post-season games. In 1997–98, he scored 9 goals and 29 points in his NHL rookie season with the Flames, earning NHL All-Rookie Team honours. He continued to improve to 34, then 38 points in his second and third seasons with the Flames before an injury-shortened season in 2000–01 kept him to 28 points in 51 games. In 2001–02, Morris was again sidelined after sustaining a left wrist injury. During the season, he was accidentally involved in a tragic incident in a game against the Columbus Blue Jackets on March 16, 2002, when an Espen Knutsen slap shot deflected off him and into the stands, striking 13-year-old Brittanie Cecil in the temple. She died from the impact two days later, prompting the NHL to implement protective netting at the ends of the rinks at the start of the following season. Morris completed the season with 34 points in 61 games. After five seasons with Calgary, reaching the 30-point mark four times, Morris was traded to the Colorado Avalanche. He made an immediate impact with the Avalanche in his first season, tallying a career-high 11 goals, 37 assists and 48 points in 2002–03. The following season, Morris was traded for the second time in as many campaigns, this time to the Phoenix Coyotes. He completed the 2003–04 season playing in 14 games and recording 2 assists with the Coyotes after being traded. He finished with a combined 32 points. In his first full season with the Coyotes in 2005–06, Morris tallied 28 points in 53 games, then recorded back-to-back 25-point seasons the following two campaigns. On March 4, 2009, Morris was again dealt at the trade deadline, this time to the New York Rangers. He did not re-sign with the Rangers in the off-season, instead signing a one-year contract with the Boston Bruins worth $3.3 million on July 24, 2009. On March 3, 2010, Morris was traded back to the Coyotes for a conditional fourth-round pick in the 2011 NHL Entry Draft. After the season, Morris signed a four-year, $11 million contract extension with the Coyotes. With Phoenix, Morris went deep into the 2012 Stanley Cup playoffs. His highlight of the playoffs was a shot from centre ice in Game 1 of the Conference Finals that beat Los Angeles Kings netminder Jonathan Quick. However, the Coyotes lost the series in five games. Though he never officially announced his retirement, Morris has not been active since the 2013–14 season.
Morris and wife Jodie have three sons: Traiten, Presley and Asher. He coached for his son Presley's team, The Jr. Coyotes 02 AA, and also for eldest son Traiten's team, The Jr. Coyotes 00 AA. Morris settled in north Scottsdale, Arizona. In the summers, he and his family return to Alberta. Morris is currently an assistant high school coach with Chaparral High.
