  We develop a mean-field theory for Bose-Einstein condensation of spin-1 atoms
with internal degrees of freedom. It is applicable to nonuniform systems at
finite temperatures with a plausible feature of satisfying the Hugenholtz-Pines
theorem and various conservation laws simultaneously. Using it, we clarify
thermodynamic properties and the excitation spectra of a uniform gas. The
condensate is confirmed to remain in the same internal state from T=0 up to
$T_{c}$ for both antiferromagnetic and ferromagnetic interactions. The
excitation spectra of the antiferromagnetic (ferromagnetic) interaction are
found to have only a single gapless mode, contrary to the prediction of the
Bogoliubov theory where three (two) of them are gapless. We present a detailed
discussion on those single-particle excitations in connection with the
collective excitations.
