Camshaft

A camshaft rotates and it makes the pistons go up and down which in turns results in the combustion cycle of an engine. As a camshaft spins, its lobes - or cams - open and close the valves on your engine in time with the piston to allow air in and exhaust out. The shape of the cam affects your engine's performance at different speeds.  Whilst there are many types of camshafts, the two most common are;

SOHC = Single Overhead Camshaft

DOHC = Double Overhead Camshaft

Pushrod = Camshaft located inside the engine rather than in the head

In a SOHC engine the camshaft is installed in the cylinder head and valves are operated either by the rocker arms or directly through the lifters

A typical DOHC engine has two camshafts and 4 valves per cylinder. One camshaft operates intake valves that are installed on one side, while another camshaft controls exhaust valves on the opposite side.

When you compare their performance you discover that SOHC engines develops a little less power than its DOHC brother, but it develops more of its power earlier and is therefore more flexible and generally more economical. The DOHCs rev much harder and higher, and at mid to top revs develop much more power for the same displacement engine, making them ideal for racing or performance vehicles. And if the DOHC is a V-engine, like a V6 or V8, it could be called a Quad-cam.
SOHC engines are cheaper and less complicated to design, build and maintain because of the lesser moving parts, while DOHC are more complex to design build and repair because of the extra moving parts.

Another advantage of having a DOHC engine over a SOHC is that the engine has twice as many intake and exhaust valves as a SOHC motor. This makes the engine run cooler and more smoothly, quietly, and efficiently. But the downfall is that DOHC engines cost more for repairs.

The whole purpose of having OHC configurations is to do away with another form of valve actuation, that of pushrods found in the Over Head Valve (OHV), which is found in most American V8 configurations (Ford uses an OHC version for their motors). OHV configurations use a centrally located cam shaft (central to the engine block), and actuates the valves through the use of lifters, pushrods, and rocker arms.

The OHC configuration does away with most of this for the reduction of mass in the valve train. This reduction of mass usually means the engine can run safer and cleaner at higher engine speeds due to less inertial pressures on the valve train. Also, since this is more of a direct actuation of the valves, there is less flex (due to push rods) and a more accurate valve actuation occurs. The trade off in this is that the belts/chains/guides which controls all of this usually wear out over a period of time well before the engine's end of life (EOL), where as on an OHV engine, the valve train and timing chain usually will last until the engine is worn out.

The OHC configuration is a lot more complex than the OHV in that there are a more parts which can go bad, which can cause catastrophic consequences if it fails. OHC engines are usually "interference motors", which means the valves and the pistons occupy the same space at different times. If the cam belt/chain fails, the valves stop moving, but the pistons do not (right away).

The piston will move up into the valves and usually destroy the valves and the entire head assembly. It is imperative that the cam belt/chain maintenance get done on time. There are no warning signs (without taking the engine apart) to tell you when it is worn out, so following the manufacturers replacement interval is very important.

As discussed, the DOHC engines usually have more valves per cylinder than the SOHC versions. They will also usually have less parts involved (most DOHC directly actuate the valves, where SOHC usually have rocker arms).

More valves means the engine can intake and exhaust more gasses from the engine. Think of a large circle, say one which is 4" across (or say 10cm for you unruly metric folks out there). How many 2" (5cm) circles can you fit within it? The answer is two. Now take the same 4" circle and fit 1.5" circles within it. The number is somewhere around four.

A 4" circle has the area of 12.57in/sq. A 2" circle has the area of 3.14 in/sq, and therefor two of those would equate to 6.28 in/sq. With the 1.5" circle, your total would be 7.07 in/sq. (Note: While I'm not sure if four could fit in the circle exactly, the example is valid for educational purposes.)

Now imagine these circles are the valves in the cylinder head. The more valves you can fit into the head, the greater the surface area you can cover. The greater surface area allows greater air/exhaust flow into and out of the cylinder. This gives the engine a chance at better volumetric efficiency or VE (basically how well the cylinder fills with the air/fuel mixture). This allows for greater performance from the engine.

Manufactures might choose SOHC over DOHC engines due to cost in design and manufacture. They might choose DOHC over SOHC for the performance aspect.

Pushrod Engines
Like SOHC and DOHC engines, the valves in a pushrod engine are located in the head, above the cylinder. The key difference is that the camshaft on a pushrod engine is inside the engine block, rather than in the head.
The cam actuates long rods that go up through the block and into the head to move the rockers. These long rods add mass to the system, which increases the load on the valve springs. This can limit the speed of pushrod engines; the overhead camshaft, which eliminates the pushrod from the system, is one of the engine technologies that made higher engine speeds possible.
The camshaft in a pushrod engine is often driven by gears or a short chain. Gear-drives are generally less prone to breakage than belt drives, which are often found in overhead cam engines.

Sources: Samarins, Engineering Explained, Nairaland, Stack Exchange, How Stuff Works