Born in Trumbull, Connecticut, Drury was chosen 72nd overall in the 1994 NHL Entry Draft by the Quebec Nordiques. After a fine career at Fairfield Prep School, Drury moved on to Boston University where he accumulated 113 goals in four seasons. He was a perennial Hockey East All-Star and All-American and was the 1998 recipient of the Hobey Baker Award as the top collegiate player in the U.S.
In 1998-99 Drury scored 20 goals for the Colorado Avalanche and was named the winner of the Calder Trophy over runner-up Marian Hossa of the Ottawa Senators. The next year he upped his point total to 67 and registered 14 points in the playoffs. Drury scored 24 goals in 2000-01 and was arguably the best third line centre in the NHL behind Joe Sakic and Peter Forsberg. When the latter went down with a season ending injury, Drury stepped up his game and scored 11 goals in the playoffs to help Colorado win its second Stanley Cup.
A silver-medalist with Team USA in Salt Lake City, Drury would spend one more season with Colorado before he was dealt to the Calgary Flames in a deal that saw Derrek Morris head to Colorado. With the Flames struggling in 2002-03, Drury was doing his best to create offensive chances and was headed for his fourth straight 40+ point season. After only one season in Calgary, Drury was dealt to the Buffalo Sabres prior to the 2003-04 season. Upon his arrival in Buffalo, Drury continued to put up solid numbers, finishing the year with 53 points on the strength of 23 goals. Even with Drury strong regular season, the Sabres failed to make the playoffs for the third straight year. Once the Sabres season came to an end, the former Hobey Baker winner was named to the U.S World Championship team and was instrumental in leading them to a bronze medal.
The following season Drury was named co-captain of the Buffalo Sabres. For the first time in Drury's tenure, the team entered into the NHL playoffs. The upstart Sabres shocked the Eastern Conference as they came within one game of entering into the Stanley Cup Finals. Earlier that season, Drury was a member of the United States Men's 2006 Winter Olympic team.
In 2006-07, Drury scored a career high 69 points co-captaining the Buffalo Sabres to an NHL-best 113 points. Expectations were at an all-time high as the President's Trophy winners entered the playoffs. However, the club failed to get past the Eastern Conference finals losing out to the Ottawa Senators.
As an unrestricted free-agent Drury signed a five-year contract with the New York Rangers in the summer of 2007 ending his tenure with the Sabres.
Though his offensive numbers dipped with the Broadway Blueshirts, Drury continued to provide exceptional two-way play for the Rangers. For his efforts he was once again selected to be part of the US Olympic team that brought home the silver medal in a thrilling overtime final at the Olympic Winter Games in Vancouver.