"The Streak" was a nickname given to Emmett Brown in his youth, as a result of his skill in sandlot football.
When running away from Kid Tannen and Matches on June 13, 1931, Marty McFly was surprised by Emmett's flawless dodging, and asked where he learned to move like that, at which point Emmett revealed that they used to call him "the Streak".
Later that evening, when Marty was trying to rescue Emmett's older, time-traveling self from a stolen police vehicle, Marty nearly fell off the side, only to be saved by Doc grabbing him through the window bars. Marty thanked Doc, noting that these quick reflexes were probably why he had been called "the Streak" in the first place. Doc questioned how Marty could possibly have known about the nickname, unaware that Marty had learned it from his younger self in this timeline.