Displacement is the size of the engine internally, measured in cubic inches (on American engines anwyay), and in CC's (cubic centemeters) on small import engines, or in liters on larger import engines.
A 350 engine means it "displaces" 350 cubic inches of "area" internally that can be filled with fuel and air. The more cubic inches an engine is, the more fuel and air it can take-in and burn to make more power.
Horsepower is a no brainer. Burn more fuel and you'll make more power. It's really not that complicated. The problem is, you MUST maintain a given air/fuel ratio for internal combustion to occur, so you can only burn just so much fuel with so much air or combustion simply won't take place or burn very well. It's a VERY narrow window in the spectrum or all air vs. all fuel. Too much fuel is like putting too many logs on a camp fire. It smothers the flame out and smokes real bad until it finally catches and burns. That takes time though. You don't have "time" inside an engine because the events happen in nano seconds.
Too little fuel and the flame is small with little heat. A small, low heat flame doesn't make for much "energy", which in an engine means low power. It really is as simple as a camp fire with gas and air inside an engine (in the overall - simple picture). How big the engine is (in displacement) is pretty much the same as how big your camp fire pit is. A bigger pit allows you the potential to burn more wood. With a large pit you can make a camp fire so big and so hot that you need to back up 20 feet or so from it to not get burned. Large displacement engines are the same way. Bigger displacement means the ability to burn more fuel, only engines convert that heat and cylinder pressure into rotating energy (horsepower).
The "magic" and complications of making horsepower in an engine is getting the engine to be low friction, low rotating mass and weight, sealing-up the cylinders yet still allowing things to move freely, combustion dynamics and physics, heat vs. expansion (because heat expands metal), etc. But in all reality, it all boils down to increasing the amount of fuel a given engine can burn to convert it into making more power. This is easiest done by increasing the cubic inches, or "displacement" of an engine.
Displacement ONLY happens because of atmosphere. If you took a 350 cubic inch engine out into space where there is no atmosphere, it would be a ZERO displacement engine because as you create the void in the cylinders, there is no atmosphere to rush-in to fill it up. On the flip side, at sea level, we live at an average of 14.7 psi of atmosphere. In other words, at any given time at sea level, we have about 14.7 pounds per square inch of "air" pushing against us. This is why if you buy a bag of tater chips at sea level and then you drive to the mountains and look at the bag, the bag is huge and bloated like it’s ready to pop. That’s because at sea level it had atmosphere pushing-in on it from all sides, and at altitude, the air is thinner and the atmospheric pressure is less, which does NOT push on the bag as much, thus allowing it to bloat and expand. This is also why when you drive to the mountains your car doesn’t run very well. It isn’t “just” because of thin air, it’s also because the atmospheric pressure is much less, so the displacement of your engine becomes much less and can’t make any power. You might have a 350 engine with 9:1 compression at sea level, but in the mountains at say, 7,000 feet, what you have is, (not actually doing exact math here), more like a 260 cubic inch engine with 7:1 compression. Again, in outer space where there is no atmosphere or pressure, a 9:1 compression engine becomes a 0:1 compression engine because there is no air or atmosphere to compress or rush-in.
Keep in mind, an engine does NOT “suck-in” air like most people think. What happens via valve timing and the piston moving up and down is inside the engine it creates a void on the intake stroke. This is what creates a vacuum inside the engine. All a vacuum is, is a void of atmosphere. As soon as the intake valve opens, atmospheric pressure PUSHES air into that void as it tries to bring it back to atmospheric pressure again. This is the same effect for why when jets at altitude have a catastrophe and a window or a door blows off, you don’t get SUCKED out of the air plane… you get PUSHED out. There is no “vacuum” at altitude to “suck” you out. At 40,000 feet the air is super thin and there is not much atmospheric pressure. This is why they PRESSURIZE the cabin of the plane. It equals out what is missing on the outside to make our bodies comfortable. If you bust out a window or a door on the plane at altitude, all of that pressurized air inside wants to escape, taking you with it. Most high flying commercial planes fly at about 8 psi of cabin pressure. That’s roughly half of atmospheric pressure. If it were true that at 40,000 feet it was roughly half of the atmospheric pressure as sea level, then at 40,000 feet a 9:1 compression engine is now making roughly half of that compression at more like 4:1 compression. An engine won’t run on 4:1, not on gas anyway. That is why your engine runs like crap at high altitudes.
So, going back to displacement. In retrospect, a 350 cubic inch engine at sea level with some form of forced induction (a supercharger, turbocharger, etc) pressurizes the intake system MORE than the atmosphere. To make this simpler, the reason a supercharged or turbocharged engine makes more power than a naturally aspirated engine is quite simple, it creates more displacement. A 350 cubic inch engine at atmosphere (14.7 psi at sea level) with a turbo or supercharger that is making between 7 and 8 psi of boost isn’t a 350 engine anymore because you have just created 50% mre pressure than atmosphere. In other words, with 7 – 8 psi of boost you just turned that little 450 into 50% mre cubic inches, so it is now a 525 cubic inch engine. Well, a 525 cubic inch engine is obviously going to make more power than a 350 cubic inch engine, right?
Let’s hone this down a bit more. If we double the atmospheric pressure on a forced induction engine, we double the cubic inches… literally. In other words, if we built a supercharged 350 engine that puts-out 14.7 psi of boost (pressure inside the intake manifold), that 350 is now “displacing” 700 cubic inches. It’s the 14.7 psi of natural atmosphere PLUS another 14.7 psi of boost from the supercharger behind it pushing fuel and air into the engine. How much power do you think a 700 cubic inch engine will make? Quite a bit, which is why a typical V-8 with a supercharger making 14 – 15 psi of boost makes a hell of a lot of power. It all boils down to displacement, which is all relevant to cubic inches and atmospheric pressure.
Always remember when dealing with engines... There's no replacement for displacement!