Mother Brain's role and physicalities are what gave her/him/it's name: it is the leader of the Space Pirates and looks like a giant brain. However, in Metroid Prime 3, a cross-section (a fancy way of saying '2D map') of a "future Aurora complex" is shown, which looks nearly identical to MB's hideout in Super Metroid. Here is a quote from Metroid Prime 3, as taken from Wikitroid:
"Twenty years ago (prior to the events of Metroid Prime 3: Corruption), a team of Federation scientists announced the birth of an organic super computer. Its name was Aurora. Since that historic date, thousands of Auroras have been built and installed. Originally designed for scientific purposes, the Auroras are now used in government, business, and military roles as well. They are given male, female, or gender-neutral personas, based on the needs and desires of the staff at the facility they are to serve in. A vast network links all existing Auroras, allowing them to access a tremendous database without peer."
"The Mother Brain was built by the Chozo, though the manga indicates that the Federation had some part in this as well, as she was built to link the Federation and Chozo information databases, similar to the function of the Auroras."