She grew up among the Vulcans after her parents were killed and attended the Vulcan Science Academy, where she studied quantum physics. Although she graduated at the top of her class, she was unable to join the Vulcan Expeditionary Group because she is human instead of Vulcan. They had to choose between letting Burnham join or letting Spock join (who is half-human), and ultimately they chose Spock over her. Sarek, Burnham’s foster father and the one who made the decision, hid his shame at making this decision by making Burnham think she was inadequate for the job. Turns out this was all for nothing, as Spock decided to join Starfleet instead of the Vulcan Expeditionary Group. However, the damage was already done and Burnham had an underlying fear that she wasn’t good enough.
Sarek then got her a job on the U.S.S. Shenzhou, where she served as a xenoanthropologist (like a regular anthropologist, but in space!) She rose to the rank of Commander before things went wrong and she ended up having to commit “mutiny” in the name of the war against the Klingons. After things went wrong on her previous ship and she ended up being arrested for the mutiny, the captain of the U.S.S. Discovery had Burnham released from her sentence early so she could come work for him and be the science specialist on the discovery and help them defeat the Klingons.