  We theoretically study electronic properties of a graphene sheet on xy plane
in a spatially nonuniform magnetic field, $B = B_0 \hat{z}$ in one domain and
$B = B_1 \hat{z}$ in the other domain, in the quantum Hall regime and in the
low-energy limit. We find that the magnetic edge states of the Dirac fermions,
formed along the boundary between the two domains, have features strongly
dependent on whether $B_0$ is parallel or antiparallel to $B_1$. In the
parallel case, when the Zeeman spin splitting can be ignored, the magnetic edge
states originating from the $n=0$ Landau levels of the two domains have
dispersionless energy levels, contrary to those from the $n \ne 0$ levels.
Here, $n$ is the graphene Landau-level index. They become dispersive as the
Zeeman splitting becomes finite or as an electrostatic step potential is
additionally applied. In the antiparallel case, the $n=0$ magnetic edge states
split into electron-like and hole-like current-carrying states. The energy gap
between the electron-like and hole-like states can be created by the Zeeman
splitting or by the step potential. These features are attributed to the fact
that the pseudo-spin of the magnetic edge states couples to the direction of
the magnetic field. We propose an Aharonov-Bohm interferometry setup in a
graphene ribbon for experimental study of the magnetic edge states.
