To elaborate a bit on engineer_comp_geek’s post, here’s my backyard mechanic’s “Internal Combustion 101 For The Total Beginner”:
Engine displacement is the amount of fuel/air mixture that can be sucked into the engine and burned on each complete combustion cycle (2 complete revolutions in the standard 4 cycle engine). This is calculated by taking the total volume traversed by one piston and multiplying by the number of pistons.
Example: Let’s say your v8 has pistons with a 4.050" bore (diameter) and a stroke (distance traveled) of 3.780". That means one piston’s total displacement is the volume of a cylinder 4.050" x 3.780" or 48.70 cu. in. So your engine would have total displacement of 48.70 x 8, which is 390 cu. in., or about 6.4L. And it would probably be a Ford. 
In other words, your normally aspirated (no turbo, etc) 390 v8 burns approximately 390 cubic inches (6.4 liters) of fuel/air mixture at the current air pressure and temperature every 2 revolutions. Burning more fuel = more power, obviously. Since a bigger engine can suck in and burn more fuel (having more displacement), it makes more power.
This is overly simplified. There are a lot of things that can affect your power. One of the biggest ones is the size of your carburetor (or fuel injector) and intake manifold. Smaller opening means less air can get through. A bigger intake is sort of like making your throttle open wider. More air and fuel can get through, so your engine burns more and makes more power.
Air moves into the cylinders at a high speed. So if it’s timed well, simple inertia will force more mixture in at slightly higher pressure, creating more power.
Turbochargers and superchargers do the same thing: they force more mixture into the cylinders mechanically (also creating higher compression), which makes even more power. The boost is measured in p.s.i. If your engine usually sucks in 2 liters at 14p.s.i. (standard pressure at sea level) and your turbo adds 8 pounds of boost, then you’re burning over 50% more fuel, and getting (in an ideal world) over 50% more power. Which is the same as having a bigger engine, but without the added weight and size.
But the old adage is true. There really is no replacement for displacement. Nothing makes more power except burning more fuel. If it weren’t so, tractor-trailer rigs and large machinery wouldn’t use large engines. Big engines with big cylinders make big torque, which is necessary for moving big things. Small engines can make a lot of power, but they are more delicate and more expensive than big ones, and don’t make as much torque. One of the main reasons they are used for racing (aside from speed limiting factors) is that smaller parts have less mass and can move faster. That means an engine can rev higher (the high pitched whine that Indy or F1 cars make, for example. Indycars hit their power band at around 10,600 rpm).
Anyway, I’ve rambled enough. Sorry for taking up so much space.