How does the number of cylinders in an engine affect its performance?

Directly, the more cylinders the more power. Let me explain:

As you probably now, in every internal combustion engine inside each cylinder there is a piston. Pistons move up-down-up (to move the whole system and finally the wheels) due to gasoline explosion, and they work in sequence. So a two cylinder engine will go 1-2-1 and so on. A four cylinder goes 1 and 3 - 2 and 4 - 1 and 3 and so on, thus in the same time double the power.

Performance-wise, when you accelerate, a more cylinders car will reach a certain speed faster. Besides, the more cylinders the more life-time an engine will have, because the pressure is shared. Of course, there are pros and cons, being the most known con your gasoline comsumption. Of course: the more cylinders the most explosions and finally the more gas.

An interesting fact is that cylinders can be "In-line" or "In-V". In line they work in a serial fashion, while in V they work in a parallel fashion. Now imagine a V12 engine (like a Ferrari Testarossa), 6 cylinders in front of another 6 cylinders, in repeated powerfull parallel sequences. This is why these cars can accelerate from 0-60 miles in just a few seconds.

Yet, you don't want to see the gas bill.

The number of cylinders affects the smoothness with which the engine delivers it's power. In a four-cycle engine, each cylinder has a power stroke once every 720 degrees of crankshaft rotation, so there is a power stroke every ( 720 / #_of_cylinders ) degrees.

Example: A Dodge Dart with a straight-six has a power stroke every ( 720 / 6 ) or 120 degrees. Four-bangers have a power stroke every (720 / 4 ) or 180 degrees.

Four-stroke engines with four or fewer cylinders have very jerky power delivery and rely heavilly on the flywheel because there are periods during the crankshaft rotation where no cylinder is producing power at all, though in a four-cylinder engine, those periods are quite brief.

Those with five or more cylinders are smoother because there is no point in time where there is not at least two cylinders somewhere in their power stroke; the engine is ALWAYS making torque and power.

Which performance parameter? In each following point, comparison assumes only the number of cylinders changes, not quality, displacement, tolerances, compression ratio or anything else, i.e. "all other things being equal."

-More cylinders means more friction surfaces (bearings, cylinder/ring contact, valve guides, etc.) and therefore lower "efficiency."
-More moving parts and more friction surfaces means more failure points, so more cylinders means lower "reliability."
-As others have pointed out, more cylinders also means smoother power delivery.
-More cylinders means more parts which means more expensive to build, buy, maintain (plugs, wires, etc.) and repair (e.g. valve job) - lower "cost performance."

In reality, all other things are never equal. The number of cylinders is a poor indicator of most performance parameters. A 2.4 liter four-cylinder may far outperform a 2.8 liter V-6 in all points - fuel economy, reliability, horsepower, torque, etc. - if it is designed and built better. A 4 liter six may likewise far outperform a 4 liter eight-cylinder. There are also plenty of situations where the contrary is true.