He was given the nickname "Bubby" (a mispronunciation of "Brother") by one of his five older sisters. Being the sixth child born was his reason for choosing 6 as his jersey number.
Brister played his high school football in Monroe, Louisiana and originally enrolled at Tulane; at that time he was known as Bubba Brister. The nickname was modified to "Bubby" sometime shortly after he transferred to Northeast Louisiana (now Louisiana-Monroe) in 1982. Brister was originally drafted to play baseball in the 4th round out of high school by the Detroit Tigers. He played one season for the minor league Bristol Tigers before attending college for football.
When Brister was selected, comparisons were almost immediately made between him and Steelers great Terry Bradshaw, who is also from Louisiana. Over the years, Pittsburgh sports writers and Steelers fans frequently made jokes about Brister's thick Southern accent and perceived lack of sophistication, traits that were similarly mocked in Bradshaw. In a similar vein, his name was often misspoken. In a 1999 Sports Illustrated article, Brister cited "Bubba Brewster" and "Bobby Blister" as common manglings.
Brister spent two years as the back up to Bradshaw's immediate successor, Mark Malone, starting two games as a rookie in 1986 and appearing briefly in relief in two games in 1987. In his NFL debut in October 1986 Pittsburgh played on Monday Night Football against rival Cincinnati Bengals and Brister passed for 191 yards and scored a rushing touchdown, although the team lost 24-22. He won a three-way competition for the Steelers' starting quarterback job with Todd Blackledge and Steve Bono.
Career highlights during his 1988-1991 run as Pittsburgh's starting QB included ranking 4th in the NFL in average yards per pass completion in 1988 and ranking 10th in the league in passer rating in 1990. Brister had five scoring passes that were 65 yards or longer in 1988, including an 89-yard touchdown to Louis Lipps vs the Philadelphia Eagles on November 13 that was the longest pass completion by a Steeler in Three Rivers Stadium history. In 1989 he set a team record with 15 consecutive pass completions in a road win over Detroit, including a 48 yarder to Lipps. Brister also set a team record in 1989 throwing 178 consecutive passes without an interception. It was 1990 that Brister established career highs for starts (16), yards passing (2,725) and touchdown passes (20). Brister missed 8 games with injuries in 1991, setting up a competition with back up Neil O'Donnell for the starting job. Pittsburgh went 5-3 when Brister played, only 2-6 with O'Donnell as a starter. Brister was the starting quarterback during Hall Of Fame Coach Chuck Noll's final post season run with the Steelers, winning the 1989 AFC Wild Card in overtime on the road against the Houston Oilers, then losing a close game to eventual AFC champion Denver Broncos. Brister led an 82-yard drive at the end of the 4th quarter to tie the Houston game and force overtime. Against Denver, he passed for 229 yards and 1 touchdown, with no turnovers.
One of Brister's famous quotes came after a 1991 game between the Houston Oilers and Pittsburgh Steelers. Pittsburgh was getting blown out, and coach Chuck Noll wanted to pull starter Neil O'Donnell and replace him with Brister to finish the hopeless game. Brister replied "I don't mop up for anybody." O'Donnell was starting in place of an injured Brister. To the surprise of many, although Brister was not forced to enter that late season loss against Houston, he did supplant the struggling O'Donnell the next week, starting the team's final two games, both wins over Cincinnati and Cleveland.
Brister played for the Steelers for 7 years, several of them as the regular starter at quarterback. In 1992, new Steelers head coach Bill Cowher chose backup quarterback Neil O'Donnell over Brister, essentially ending his career as a starting player for the Steelers. Still, Brister played a significant role in the team's 1992 success. Brister won two games as a starter for an injured O'Donnell against the Indianapolis Colts and Cleveland Browns. In the Cleveland game, the Steelers needed to win to clinch home field advantage throughout the AFC Playoffs and Brister passed for 223 yards, 1 touchdown, no interceptions, and had a string of 11 consecutive pass completions in one stretch. In two other games Brister came off the bench, relieving a struggling O'Donnell when he was hurt and lead 4th quarter comebacks over the Seattle Seahawks and Detroit Lions.
After brief stops as a backup quarterback for the Eagles and New York Jets, Brister sat out the 1996 season. In his first season in Philadelphia in 1993, Brister ranked 7th in the league in passer rating and 4th in lowest interception percentage, starting 8 games with two relief appearances subbing for an injured Randall Cunningham. Highlights that season included his 27 completion, 245 yard, two touchdown performance vs. the Super Bowl champion Dallas Cowboys on Monday Night Football, and his 353-yard, three touchdown performance in the final game of the regular season, again on Monday Night Football, leading the Eagles to a win over NFC West champion San Francisco. Brister's interception percentage that year was the lowest in Eagles team history for more than a decade until eclipsed by Donovan McNabb. His former teammate in Philadelphia, linebacker Bill Romanowski, had signed with the Denver Broncos and, in 1997, suggested to Broncos head coach Mike Shanahan that he take a look at Brister. Brister signed with Denver and became their #3 quarterback for the 1997 season, backing up Hall of Fame quarterback John Elway and Jeff Lewis. Brister's only significant playing time in the 1997 season came in a week 17 game against the San Diego Chargers. Before the 1998 season, Jeff Lewis had fallen out of favor with the Broncos; thus, Brister became the primary backup QB. During the 1998 season, Elway was forced to sit out a number of games due to injury and Brister started those games in his place. He played well and the Broncos went undefeated in all of his starts (4-0); Brister also broke the team's (then) record for longest rushing touchdown by a quarterback and recorded a higher passer rating than Elway. However, when Elway retired in 1999, Brister was passed over for the starting spot in favor of Brian Griese, and the Broncos released him after that season. During his three seasons with the Broncos, he won two Super Bowl rings.
Brister spent 2000 with the Vikings. He signed with the Kansas City Chiefs in 2001 but was cut before the season began. He then retired from football. He finished his career with a passing record of 1,207 completions in 2,212 attempts for 14,445 passing yards and 81 touchdowns.
Brister played in three conference championship games and two Super Bowls.
After retiring from football, Brister spent a short time as a television sports analyst for Fox Sports Rocky Mountain in Denver. In 2003, Brister became the co-host of a hunting and fishing oriented show called Louisiana Outdoor Adventures on The Outdoor Channel. In 2005, he joined the staff of Hunter’s Specialties, a producer of hunting and fishing adventure videos.