Weber is predominantly known for his powerful slapshot, winning the hardest shot event four times at the annual NHL All-Star Skills Competition.

He also scored 5 points in 19 playoff games as the Rockets won the franchise's first Ed Chynoweth Cup as WHL champions, defeating the Red Deer Rebels in the finals.

[14] In his second season with the Rockets, Weber was named to the WHL roster for the 2004 ADT Canada-Russia Challenge in Red Deer and Lethbridge, Alberta.

[20] The injury resulted in Weber missing the remainder of the regular season and the Rockets' Western Conference quarter-finals series against the Vancouver Giants.

[20][21] Despite the injury, Weber finished the regular season with WHL career bests for goals (12), assists (29) and points (41) in 55 games played.

[26][27] A year-and-a-half later, Weber made his NHL debut on January 6, 2006, against the Detroit Red Wings, recording three shots on goal in 11:08 minutes of icetime.

He also scored two goals in four Stanley Cup playoff games with the Predators before the team was eliminated in the first round by the San Jose Sharks.

[34] By the midpoint of the season, Weber already had 26 points,[34] and his play was recognized on January 14, 2007, when he was named to the Western Conference roster for the 2007 NHL YoungStars Game in Dallas, Texas.

After playing only 2:19 minutes of the first period in Nashville's season-opening game against the Colorado Avalanche, Weber fell awkwardly and dislocated his kneecap.

[41] On June 23, 2008, Weber signed a three-year, $13.5 million contract extension with the Predators, avoiding restricted free agency.

At the mid-way point of the season, he was among the defencemen statistical leaders, was considered a favourite to win the Norris Trophy and was named to the Western Conference roster for the 2009 National Hockey League All-Star Game in Montreal.

[43][44] Weber finished the season with career-highs in all major statistical categories, including games played (81), goals (23), assists (30), points (53) and penalty minutes (80).

[49] He also established a strong leadership role and willingness to defend his teammates, highlighted by three consecutive games in March 2010 in which he fought opposing players.

[51] On July 8, 2010, Weber was named the fifth captain in Nashville Predators history, replacing Jason Arnott, who was traded to the New Jersey Devils.

[67] Already having lost Weber's defensive partner Ryan Suter to unrestricted free agency, the Predators matched the offer sheet five days later.

During the 2015–16 season, Weber recorded his first career hat-trick on December 5 against the Detroit Red Wings, which was also the first-ever by a Predators defenceman.

[75] On October 1, 2018, Weber was named the 30th captain of the Canadiens, replacing Max Pacioretty after he was traded to the Vegas Golden Knights on September 10.

Even in those circumstances, he led the team's top defensive line and on-ice time, guiding them to a surprisingly deep playoff run that took them all the way to the Stanley Cup Finals.

In the 2021 off-season, Montreal's front office announced that the physical toll of Weber's injuries was too significant for him to overcome and that he would not be playing in the 2021–22 season, with the possibility of early retirement despite five years remaining on his contract.

The Golden Knights acquired Weber's contract for purposes of salary cap relief, and there was not an expectation that he would play for the team.

[81][82] On February 22, 2023, the Golden Knights traded Weber's contract and a fifth-round pick in 2023 to the Arizona Coyotes in exchange for Dysin Mayo.

[83] Weber's contract rights, along with the other assets of the Coyotes, were acquired by the Utah Hockey Club on April 18, 2024, when the NHL Board of Governors approved an expansion franchise for Salt Lake City beginning in 2024–25.

[85][86] There, his play with Dion Phaneuf formed the team's top defensive pair, helping Canada win its first junior gold medal since the 1997 tournament.

[88] Weber debuted with the national senior team on April 22, 2007, accepting an invitation to join the club for the 2007 IIHF World Championship in Russia.

[90] One minute into Canada's preliminary round game against Germany, Weber hit Yannic Seidenberg in the chin with his elbow, giving the German a concussion and sidelining him for the remainder of the tournament.

[91] After serving his suspension, Weber continued with the tournament, finishing with two points in six games, including a goal in Canada's 5–1 quarter-final victory over Switzerland.

[93] On April 14, 2009, Weber was named to the national senior team for the 2009 IIHF World Championship in Switzerland,[94] as one of four alternate captains.

His shot passed through the mesh net behind German goaltender Thomas Greiss and a video review was required to award the goal.

[106] After his Nashville Predators failed to make the 2013 Stanley Cup playoffs, Weber was considered a lock for the 2013 World Championship team, but Hockey Canada deemed the cost of insuring his large NHL contract too high and ultimately did not select him.

When Weber was 14, his mother underwent surgery to remove brain tumors, but in early 2010 had a series of seizures, was placed in an induced coma, and died on August 11, 2010.