As a youth, Bradley played in the 1978 Quebec International Pee-Wee Hockey Tournament with a minor ice hockey team from Kitchener, Ontario. Brian Bradley's NHL debut came in the 1985–86 season with the Calgary Flames, where he saw very limited action, although his first playoff game was during the 1986 Stanley Cup Finals against the Montreal Canadiens. He would spend most of the season with the Moncton Golden Flames, where he was linemates with future superstar Brett Hull. In 1988, after arriving back from playing with Canadian National Men's Hockey Team, where he spent most of the 1986–87 NHL season playing, Bradley was traded to the Vancouver Canucks. During the 1989 playoffs, Bradley would tie rookie Trevor Linden with a team-leading 7 points in seven games. His best regular season totals with the Canucks came in the 1989–90 season when he scored a team respectable 48 points and was awarded The Canucks' "Most Exciting Player Award" by Canuck fans. He started out the 1990–91 season playing strongly, only to be traded to the Toronto Maple Leafs for mobile defenceman Tom Kurvers. In 1992, the expansion Tampa Bay Lightning acquired him in the 1992 NHL Expansion Draft, and he would become the team's first star. He would score the team's first preseason goal against the Minnesota North Stars. He would set personal highs in goals, assists and points before the All-Star Break. At the end of Tampa Bay's inaugural season, Bradley led the team with a career high 42 goals and 86 points. He played well enough throughout the season that he also made his NHL All-Star debut. In 1995–96 he set a personal high of 56 assists. That same year the Lightning debuted in the NHL playoffs. The next season Brian Bradley would score the first goal in the history of the Ice Palace arena. Unfortunately, he would be sidelined for most of the rest of the season due to a concussion. He would play with the Lightning until retiring due to chronic injuries on December 23, 1999.
Post-hockey life
Bradley is still affiliated with the Lightning's organization and makes frequent appearances on Sun Sports television, which broadcasts regular season Lightning games. On March 29, 2017, Bradley was inducted into the Sports Club of Tampa BayHall of Fame as part of the Hall's 34th class. Bradley became the third member of the Lightning organization to be inducted. This is Bradley's second hall of fame induction in that he is also a member of the Waterloo Region Hall of Fame in his hometown of Kitchener, Ontario.
