The Cabildo was the center of the Spanish Government when Argentina was under colonial rule. The original adobe structure was built in 1580, and the white stucco building you see dates back to 1810.
Every major city had a Cablido on the main square which served as the seat of the local government when Argentina was under Spanish rule. The one you see here used to be larger but portions on each side were demolished to create the avenues you see bordering the Cabildo.
Inside is a history museum and I find it really worthwhile. Entrance is free of charge, so you have no excuse.
Whether you take a while reading the plaques or fly right through, make sure you head up to the second story balcony for a great view of Plaza de Mayo and the Casa Rosada.