We shall keep it simple, and only include include engines that you can run on derivetives of crude oil:
SI IC engines (petrol, paraffin, LPG)
CI IC engines (diesel)
Jet engines
Both IC engines can be piston engines or rotary (wankel).
The most common car engine is a flat four 4stroke SI IC engine (petrol). Four cylinders are lined up and attached to a crankshaft. For every two turns of the crankshaft (or four cylinder strokes), each piston fires once.
Also common in cars are flat four 4stroke CI IC engines (diesel engines).
Other types of and layouts for cylinder engines include:
1cyl 2/ (small motorcycles and scooters)
flat3 4/ (aygo)
HO4 4/ (VW beetles and campers)
flat6 4/ (from here on in, we're talking super cars and large machinery)
V6 4/
V8 4/
V10 4/
V12 4/
Radial 4/ (prop driven aircraft)
A 2/ (two-stroke) fires once for every turn of the crank shaft.
A flat or in-line engine has all of the cylinders in a row
An HO (horizontally opposed) engine has cylinders on both sides of the crankshaft, exactly opposite each other.
A V engine has two rows of cylinders in a V-shape
A radial engine has the cylinders arranged in a radial pattern around the crankshaft
Also made by mazda are the wankel engines. These have no cylinders, but a rotating central triangular structure instead. They achieve much higher revs and a higher power/weight value.
Jet engines are only really used on aircraft, as they are difficult to attach to a propshaft.