How do turbochargers work?

A turbocharger converts exhaust enthalpy (temperature, flow, & pressure) from a turbine into rotational shaft-work yielding into increased intake manifold pressure through spinning a centrifugal compressor. 

A typical exhaust turbocharger, exhaust enthalpy (temperature, flow, & pressure) is converted into increased air pressure.

Stoichiometric combustion, or "stoich", occurs at an air:fuel (AFR) mass ratio of 14.7:1.  This means that for every 1 gram of fuel, 14.7 grams of air are necessary for complete combustion.  Any less than 14.7 grams of air and the 1 gram of fuel will not fully combust (i.e. fuel-rich); any more than 14.7 grams and there is excess air in the engine (i.e. fuel-lean).  We're going to gloss over the "how, why, when, what" of fuel-rich versus fuel-lean combustion scenarios at this time, and simply state that peak power occurs at stoichiometric combustion (AFR 14.7:1).

AFR: 14.7:1 is the Goldilocks zone for gasoline engines

Okay great, so what does AFR have to do with turbocharging?  Well, everything. 

Say we have a naturally-aspirated (i.e. breathing atmospheric pressure, at 1 bar (14.7 psia)) 4 cylinder 2.0L engine which outputs 150 horsepower.  What if we ran this engine on some faraway "Planet  X" where the air was twice as dense, or 2 bar (29.4 psia) pressure?  The engine on Planet X would ingest twice (2x) as much air as on Planet Earth.  Effectively turning the 2.0L Planet X engine into a 4.0L Earth engine.  Going back to stoichiometric combustion (AFR 14.7:1), that means we can now burn twice (2x) as much fuel resulting in twice (2x) the power while maintaining the air-fuel ratio (2*14.7 air:2*1 fuel) = (14.7 air: 1 fuel).

Now, that is assuming i) no other losses from pumping/compession, ii) the fuel system could support 2x additional fuel flow, & iii) the mechanical strength of the engine could support it (e.g. higher in-cylinder pressure and head gasket blow-out, piston damage, or bent connecting rods).  All of that aside, this engine would make double the power! 

Our 2.0L 4 cylinder engine now thinks it is twice as large (4.0L) when boosted to 2 bar (29.4 psia) IMP.  Here, a sequential turbo setup is shown.

Back on Earth, if we attach a turbocharger to this engine, allowing for boosted intake manifold pressure (IMP) to 2 bar (29.4 psia), the engine can make double the power.  Again, assuming no other losses.  Aren't turbos great?

--Axle