Not that the subject hasn't been thrashed to within 25.4mm of its life, but to summarize:
1) Simplest is an alternator providing AC output - probably 3 phase, frequency varying with the engine rpm (wild frequency). (May be called a "generator" in defiance of most norms)
2) Said alternator may be packaged with a set of solid-state diodes, which convert AC to a kind of sloppy DC. Typically employed in automobiles for about the last 50 years, and more recently in light aircraft.
3) If a commutator and brushes is packaged in the case to convert the AC to DC, it's a traditional generator from nearly a century ago.
4) If fixed-frequency AC is required, as in transports starting from the 50s, a variable-ratio hydromechanical drive (CSD = constant-speed drive) may be employed driving the 1) alternator. If the CSD and alternator are packaged as a single unit, it's called an IDG (integrated drive generator).