The Brown Institute For Social Science, commonly referred to simply as the Brown Institute, was a research laboratory for developing scientific inventions that had social applications. It existed in the timeline where Emmett and Edna Brown, as First and Second Citizen Brown, ran Hill Valley as a gated city with strict rules.
It was converted from the Brown family mansion, as Emmett Brown was never interested in time travel in this timeline. Therefore, the temporal field capacitor was never invented, so the fire that burned down the mansion in 1962 never occurred.
Second Citizen Brown had the DeLorean time machine taken here. When Marty and Citizen Brown escaped the Citizen Plus ward, Citizen Brown told Joey Baines to guard the institute, telling him not to let anyone in. Citizen Brown then went inside to fix the DeLorean, but had Marty wait outside so his presence in fixing the DeLorean wouldn't disrupt the space time continuum.
Second Citizen Brown shocked Joey with a taser, and told a hypnotized Biff Tannen to set the timer on the dynamite to kill her husband and Marty, making it look like Marty bombed the institute. Luckily, Citizen Brown's plan worked, and he arrived in the DeLorean from six months in the future. Citizen Brown and Marty returned to 1931 to prevent Emmett from marrying Edna, and the bomb blew up in Biff's hands as Edna never told him what to do with it after he set the timer.