The Fermi surface (FS) is the surface on reciprocal space which separates occupied from unoccupied electron states at zero temperature.
The importance of FS is that the electrons at (or within an energy \(\sim k_BT\) of) the FS are special because Pauli exclusion principle preventing electrons from deep in the Fermi sea from being easily excited (and thus making any contribution to the transport properties).
The dynamical properties of an electron on the FS largely depend on its position on the FS, and the shape of the FS with respect to the Brillouin zone can be a guide to the electrical properties of the metal.
The crystalline state is not isotropic, and the presence of a finite potential (from e-ion and e-e interaction) modifies the shape of the bands so that their energies no longer have to depend quadratically on the wave vector, and the FD can distort from the free-electron sphere. i.e. FS of real metals are not spherical