Vehicle owner's manuals give motor oil recommendations based on what works best with the engines made by the company.  Choice of viscosity grades is usually provided depending on ambient temperature conditions.

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

5W-30 is now approved for most late model engines on a year-round basis.  It is not approved for many turbocharged or diesel applications, some high output V-8s, or applications that involve driving at sustained highway speeds or towing in hot weather.  It may not be the best choice for older engines with high mileage.
5W-30 is the factory fill oil on most new cars because it pumps through the engine more quickly after start-up.  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.  Even so, some people prefer them.  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 is too thick at low temperatures for easy cold starting)

Special multiviscosity oils, like 20W-50, are typically formulated for racing or severe duty applications such as towing.  They are not usually intended for everyday driving.

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