What type of motor oil is recommended for my car engine

Use the type of motor oil specified in your customer's owners manual. Most manuals say it's okay to use a variety of viscosity grades depending on temperature conditions. Generally speaking, the following holds true:

10W-30 is best for all engines for year-round driving. 10W-40 is more popular in the aftermarket, but 10W-30 is actually a better oil because the additive package in it holds up better over the long haul. This is why General Motors does not recommend 10W-40 motor oils for any of its cars.

5W-30 is approved for most late-model four-cylinder, V-6 and V-8 engines on a year-round basis. It is not approved for many turbocharged or diesel applications, some high output V-8s, or applications involving driving at sustained highway speeds or towing in hot weather. It may not be the best choice for older, high mileage engines.

5W-30 is used as the factory fill oil on most new cars because it pumps through the engine more quickly after start-up (important for keeping overhead cams properly lubed). It also makes cold weather starting easier and reduces fuel consumption.

Straight viscosity oils have limited temperature ranges and lack the versatility of multiviscosity oils. They can be safely used as long as their temperature limits are observed.
Straight 10W is okay for cold weather starting and driving, but too thin for warm weather driving.
Straight 20W is okay for all around driving, but doesn't provide the temperature protection of straight 30W (which gets too thick at low temperatures for easy cold starting).
Straight 40W and 50W oils are primarily for heavy-duty applications.

Special multiviscosity oils such as 2OW-50 are typically formulated for racing or severe duty applications such as towing.

Synthetics are a good alternative for any of the above because most provide extended temperature protection and service life.