The basic operation of the four-stroke internal combustion engine is; suck, squeeze, bang, blow. At least that's an easy way for beginners to remember it.
The first cycle is called induction (or intake) where it draws in the air and fuel. This roughly coincides with the piston going down (which is only part of the reason that air and fuel get drawn in, but we'll keep this simple).
For much of the time that the piston goes back up, the engine proceeds through the second cycle where it compresses that air and fuel mixture ready for combustion, because more energy (power) can be extracted from burning that mixture when it has been compressed (but there's a limit to how much it can be compressed before things go wrong).
The third stroke is the combustion event (the power stroke). When the piston goes down this time it is transferring energy (via the connecting rod) into the crankshaft to keep it spinning (often with the goal to make it spin faster). And the final stroke allows the exhaust to be expelled (which it actually does of its own accord when the valve opens due to the high pressure it is under; the piston does almost nothing to push it out).
As such, they can also be thought of as a pump, drawing gas and probably some liquid in, then expelling gas and particles and a little bit of unburnt liquid too. We can also assist them with another pump upstream of that induction cycle. We call this forced induction.
Forced induction has been around for a very long time. Cars started using it in the 1920s. Piston-powered aircraft in WWII used it to reach higher altitudes than they could without forced induction. There are two common ways to force the induction. One is a superchanger, and there are multiple variations of those. And the other way is to use one or more turbochargers (which can be configured in many ways).
Different forced induction setups will have different power delivery though. A supercharger is connected to the crankshaft, often by a belt or chain. That rotates the internal workings of the supercharger to pump more intake mixture into the engine during the induction stroke. This takes a bit of power away at the crank, but it more than pays it back so there's a net increase in power. The benefit of a supercharger over a turbocharger is that a supercharger's operation is tied to engine rpm, so it's very predictable for the driver.
They're not all the same though. A positive displacement supercharger (usually one of the types that you see bolted directly atop a V8 for instance) will pump in an approximately proportional amount relative to the engine. What that means is, when we measure the additional intake pressure (often called boost pressure) with the throttle wide open, that pressure reading (theoretically) remains the same through the rev range (other factors come into play though).
A centrifugal supercharger is basically a belt-driven turbine, and up to a certain point turbines work much more efficiently as they spin faster. That means the boost pressure rises as the revs rise. It's still pretty predictable though, it's just a much steeper power curve than most alternative power units.
Turbochargers have two turbine wheels (in separate housings) that share a common shaft. One turbine is harnessing some of the exhaust energy on its way out to the atmosphere, and the other turbine is pumping more induction mixture into the engine. In automotive applications they typically need to spin somewhere between 80,000 to 120,000rpm to work effectively.
The advantage of the turbocharger is it only inhibits the expulsion of exhaust to a small degree compared to the energy taken directly from the crankshaft for the supercharger. That means, for a given boost pressure and rpm, the turbocharged engine will put out a bit more power than an otherwise identical one with a supercharger.
The disadvantage of the turbocharger is, if there's not enough exhaust being produced to get them up to their operating speed, then there's no forced induction happening. That delay between squeezing the throttle and getting boost pressure is called lag. Bigger turbos can help make more power, but need more exhaust to energise them up to enough rpm to do their job. Smaller turbos will offer less peak power, but also have less lag.