1. Series, where it is fuel -> engine -> battery -> e-motor -> wheels , with regenerative breaking as a second power source to the batteries. These are range-extended vehicles, where a small generator allows both for more range than the small batteries could supply, and the ability to run without electric charging.

2. Parallel, where both systems are driving the wheels through a clutch system. Both systems would be expected to operate at the same rate if both are engaged. This is a motor-assist system, and the majority of road power is supplied by the gasoline engine.

3. Series/Parallel (or Split) where the two systems are mixed using a planetary gear system. Among other things, this allows for gasoline motors to be optimized for efficiency with a specific set of operating conditions, knowing the electric motor can smooth out operation.

4. Plug ins, very similar to series/parallel but where the wheels are now being driven primarily by the electric motor. The e-motor is more powerful and there is significantly more battery, which now can be charged independently. The batteries are still far below a full EV, having only 5-10% of the range available as a "full EV" mode.

All of these systems effectively have a single point of interaction between the gasoline and electric systems - for series it is the batteries, for parallel it is the clutch, for split systems and plug-ins it is the planetary gear system. The difference I suppose is that the series system has a wire connection to the gasoline engine rather than a physical drive connection, so conceptually it seems easier to swap out.

I heard that the Chevy Volt as an example was originally planned to be a series system, but the electric motor simply wasn't capable of providing the highway performance they wanted - so it ultimately shipped as a series-parallel system.

Hybrid vehicles have a transmission to drive the wheels from both ICE and electric.