Laird attended La Quinta High School in Westminster, California. He was drafted by the Cleveland Indians in the 27th round of the 2005 Major League Baseball draft. He did not sign, opting to attend Cypress College, where he played college baseball for the Cypress Chargers.
The New York Yankees selected Laird in the 27th round of the 2007 Major League Baseball draft. He signed with the Yankees.
On August 2, 2010, Laird was promoted from the Trenton Thunder of the Class AA Eastern League to the Scranton/Wilkes-Barre Yankees of the Class AAA International League. Laird was named both the Most Valuable Player and Rookie of the Year of the Eastern League in 2010, following a breakout season. Following the 2010 season, he was added to the Yankees' 40 man roster in order to protect him from the Rule 5 Draft.
Laird was promoted to the majors for the first time on July 18, 2011, when Ramiro Peña was placed on the disabled list. On July 22, at Yankee Stadium, Laird made his major league debut, and got his first major league hit and RBI against Oakland Athletics pitcher Craig Breslow on a single to center field. He scored his first run earlier in the game, scoring on a single by Nick Swisher after working a walk off of A's pitcher Joey Devine. The Yankees designated Laird for assignment on August 27, 2012.
Laird was claimed by the Houston Astros on September 1, 2012. Laird started the 2013 season with the Triple-A Oklahoma City RedHawks. The Astros promoted Laird to the big leagues on April 18, 2013, after Brett Wallace was optioned to Oklahoma City. Laird had been hitting .353 in 12 games to earn his promotion. He was outrighted off the roster on October 2, 2013. Laird ended his 2013 season batting .169.
Laird signed a minor league deal with the Kansas City Royals on November 23, 2013. On March 15, 2014, Laird was traded to the Washington Nationals for a player to be named later. He played for the Syracuse Chiefs of the International League, and was named the league's player of the week for the week of June 16‐22.
In November 2014, Laird signed with the Hokkaido Nippon-Ham Fighters of Japan's Nippon Professional Baseball. Laird struggled to start the 2015 season with Hokkaido, batting under .200 into June, but finished his first season with 34 home runs and a batting average of .231. In 2016, Laird led the Pacific League with 39 home runs. He hit the game-winning home run in Game 4 of the 2016 Japan Series, and a grand slam in Game 6. Laird won the Japan Series Most Valuable Player Award.
Brandon's brother, Gerald, is a catcher who most recently played in the Majors in 2015 for the Arizona Diamondbacks and is currently manager of the Single-A Detroit Tigers affiliate, the Connecticut Tigers of the New York–Penn League.