What is Courts Of Westminster Hall?

Legal Definition
The superior courts, both of law and equity, were for centuries fixed at Westminster, an ancient palace of the monarchs of England. Formerly, all the superior courts were held before the king's capital justiciary of England, in the aula regis, or such of his palaces wherein his royal person resided, and removed with his household from one end of the kingdom to another. This was found to occasion great inconvenience to the suitors, to remedy which it was made an article of the great charter of liberties, both of King John and King Henry III., that "common pleas should no longer follow the king's court, but be held in some certain place," in consequence of which they have ever since been held (a few necessary removals in times of the plague excepted) in the palace of Westminster only.

The courts of equity also sit at Westminster, nominally, during term time, although, actually, only during the first day of term, for they generally sit in courts provided for the purpose in, or in the neighborhood of, Lincoln's Inn. Brown.
-- Black's Law Dictionary