There are many different types of rocket engines, usually referred to as rocket engine cycles. We will be discussing 3 main cycles in this blog post. Open Cycle/Gas Generator, Closed Cycle/Staged Combustion Cycle, Expander cycle, and pressure-fed cycle.
Open Cycle/Gas Generator Cycle:
The open cycle class rocket engines are engines that take a tiny bit of fuel and a tiny bit of oxidizer, and turn it into a gas by burning it, to spin the turbopumps. That small amount of fuel and oxidizer is being burned in the gas generator chamber. The turbopumps are the super fast spinning pumps that convert low pressure fuel to high pressure, shooting them into the combustion chamber. The exhaust from gas generator chamber is then dumped overboard from a pipe. This leads to a waste of efficiency and slightly less thrust.
Closed Cycle/Staged Combustion Cycle:
The closed cycle class rocket engines are nearly the same as open cycle, except instead of dumping the gas generator exhaust overboard on another pipe, it is then re-routed back into the main fuel/oxidizer combustion chamber, giving it the name, closed cycle, because you close the cycle. This is why the gas generator is renamed to a preburner, as it preburns the fuel and doesn't dump any generated exhaust gases overboard. This gives you more efficiency and potentially slightly more thrust if made correctly. Examples of Closed Cycle/Staged Combustion Cycle engines are, the Aerojet Rocketdyne RS-25 Engine, the Blue Origin BE-4 Engine, and the SpaceX Raptor Engine. Although Closed Cycle is good and more efficient than open cycle, it is often way more expensive and complicated.
The expander cycle is similar to both open cycle and closed cycle. Usually, the expander cycle is more like the closed cycle engines, as all the fuel ends up flowing through the main combustion chamber. What this cycle does is, it cools the nozzle by flowing cryogenic fluids like Liquid Oxygen, Liquid Hydrogen, or Liquid Methane through the nozzle and chamber walls, and then since that extremely cold liquid heats up in the nozzle walls from the extremely hot combustion happening inside the nozzle, the cryogenic liquid in the nozzle wall heats up and expands into a gas. The gas is then re-routed to spin the main turbopumps, and then that gas is the re-routed into the main combustion chamber, as it would be pointless and potentially dangerous to dump it overboard. It also wouldn't really be possible to "dump" a gas overboard.
The pressure fed rocket engine cycle is usually the least powerful. There is no gas generator exhaust pipe to dump the gas generator exhaust overboard, and no gas generator/preburner exhaust is re-routed into the main combustion chamber, because there is no gas generator or preburner, as this cycle does not have any turbopumps/pumps at all. This cycle literally uses pressure in the fuel and oxidizer tank to push the fuel and oxidizer into the fuel injector to then be sprayed into the main combustion chamber. A major disadvantage in this cycle is, the tank pressure always needs to be higher than the chamber pressure where fuel and oxidizer is constantly combusting creating a crazy amount of pressure, and usually rocket engine tanks are not at a crazy high pressure, as the turbopumps make up for it by converting the fuel and oxidizer from low pressure to high pressure. You also have less throttle control, as all you have to control throttle is the valves, to partially open and close them, while you can control turbopumps more by simply adjusting the speed of the turbopump.
Out of all 4 of these cycles, the most efficient is between the expander cycle and closed cycle. Closed Cycle is most likely the better option for thrust and efficiency, but those 2 cycles are most likely the best options, unless you are on a budget. Most amateur aerospace engineers and rocket scientists use the pressure fed cycle, as it is simple, cheap, and the parts are usually globally accessible.