Rockets are powered by injecting a fuel and an oxidizer together into a combustion chamber. You may recall from chemistry class, oxidizers are compounds that cause other compounds to lose their electrons. Since a chemical reaction is nothing more than an exchange of electrons, you need an oxidizer to make a reaction happen. In everyday combustion reactions (like burning a match), the oxidizer is readily available in the air as oxygen. But in space, you need to carry your own. 

To start, there are a few concepts we need to lay down. First, mass flow is conserved as the fluid flows through the nozzle. You wouldn't expect fuel to enter the nozzle at 5kg/second and exit at 3kg/second. Second, there are three factors that determine mass flow rate through a duct: 1) fluid density, 2) cross-sectional area of the duct, and 3) fluid velocity. And finally, our goal is to expel the exhaust at the highest velocity possible in order to maximize thrust

For the converging section, the more we decrease the cross-sectional width of the nozzle, the faster the fluid must go to maintain conservation of mass. This makes sense. But if you think about the diverging section (once the fluid has passed the throat), you may notice something doesn't make intuitive sense. If decreasing width increases exhaust speed, why would we ever widen the nozzle? Shouldn't that slow the exhaust down? How does it continue to speed up?

The answer has to do with the compressibility of fluids. At low speeds, fluids don't compress very much, so reductions in cross-sectional area lead to increases in velocity. But at some point, decreasing cross-sectional area no longer adds more velocity, it starts increasing the density of the exhaust, which isn't helpful. This happens right as the speed of sound is hit, so at this point, gains in velocity are made by reducing density, hence the increasing area. Therefore, the throat has to be put right where the exhaust surpasses the speed of sound. 

In short, the reason for rocket nozzle shapes has to do with inherently difference characteristics of subsonic and supersonic fluids. For more, I recommend this link and this video. 

Getting the fuel and the oxidizer into the combustion chamber at such high pressures is a huge challenge and requires a lot of pumps and valves form a combustion cycle. There are in fact a number of designs, and the one selected depends on the requirements of a given application (efficiency vs. simplicity, cost vs. reliability, etc). I'll discuss three main designs

• Pressure-fed cycle: An inert gas (helium) is used to force the fuel and oxidizer into the combustion chamber. Simplest and cheapest, but most inefficient due to the added weight of the inert gas tank and the presence of an inert gas in the combustion chamber

• Gas-generator cycle (open cycle): A small portion of fuel and oxidizer is diverted into a small pre-burner. The exhaust from the pre-burner is used to power a turbopump that draws the fuel and oxidizer into the main combustion chamber. This method is also relatively simple and cheap, but the main inefficiency is that the exhaust from the pre-burner is expelled and wasted