  In this paper we study Dirac-Hestenes spinor fields (DHSF) on a
four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields
must be defined as certain equivalence classes of even sections of the Clifford
bundle (over the RCST), thereby being certain particular sections of a new
bundle named Spin-Clifford bundle (SCB). The conditions for the existence of
the SCB are studied and are shown to be equivalent to the famous Geroch's
theorem concerning to the existence of spinor structures in a Lorentzian
spacetime. We introduce also the covariant and algebraic Dirac spinor fields
and compare these with DHSF, showing that all the three kinds of spinor fields
contain the same mathematical and physical information. We clarify also the
notion of (Crumeyrolle's) amorphous spinors (Dirac-K\"ahler spinor fields are
of this type), showing that they cannot be used to describe fermionic fields.
We develop a rigorous theory for the covariant derivatives of Clifford fields
(sections of the Clifford bundle (CB)) and of Dirac-Hestenes spinor fields. We
show how to generalize the original Dirac-Hestenes equation in Minkowski
spacetime for the case of a RCST. Our results are obtained from a variational
principle formulated through the multiform derivative approach to Lagrangian
field theory in the Clifford bundle.
