For the AC generator, that is the end of the electric story except that the magnet is the moving part and the coils are fixed. The north and south poles of the magnet will cause the current to flow in opposite directions in any wire they pass, thus causing an "alternating Current."
The difference for a DC generator is addition of a commutator and the field is fixed while the coils rotate. Each coil is attached to two commutator "bars" at the end of the shaft as shown in the attached figure. The "brushes" (E on the figure) are two, electrically conducting, solid carbon blocks [shown as six white blocks in the (b) figure] that slide over the commutator bars as the center "armature" rotates. The bars and the commutator form mechanical rectifier system so that the current flow is always out at one brush and in at the other no matter what the position of the rotor.
This is the original design for a DC generator. With the advent of the transistor, the modern day "alternator" used in automobiles is an AC generator where silicon rectifiers that only allow current to flow in one direction are used to convert the AC to DC to charge the battery. Through much less efficient than the previously used automotive generators, they require less maintenance, and that is a big sales item in today's car market.
Don Howard