Its easy to misunderstand oil viscosity ratings, just as its easy to misunderstand everything else about motor oil. Without sounding too much like a conspiracy theorist, one might say that this confusion plays right into oil makers' hands - but that's another post.
As Hondaman explains, multi-grade oils are designed with a thin base oil to which special polymers are added. What he fails to mention is that these polymers thicken the oil as the oil gets hotter and only as the oil gets hotter. In other words, at 20o the oil's viscosity is described by the first number (just as @charlevoix418 mentions) and at 210o the viscosity is described by the second number (As Hondaman mentions). For temperatures in between, the oils viscosity is somewhere in between.
Q:If the polymers were not added, what would happen? A:The oil's viscosity would steadily drop as the oil got hotter. Unfortunately, this thinning is the opposite of what an engine needs as it operates under load at full temperature. But, polymers fix that, e.g. an SAE30 oil is thicker at 20o then at 210o, but a 5W30 oil is thinner at 20o then at 210o, and is the same viscosity as the SAE30 oil at 210o.
Another reason viscosity reversal is important - At start-up we want the oil to begin lubricating the parts as quickly as possible, and thinner oil does that. If we tried using an SAE30 oil at start-up, it would be too slow to lubricate that all important camshaft. If we tried using an SAE10 oil, it would be great at start-up, but then at operating temperatures would be too thin to work. Polymers fix that too.
Also, as the temperature drops further, then even SAE10 oil would become too thick to quickly lube that camshaft at start-up, so we must use an even thinner oil - SAE5 or even SAE0. Hence, as the temperature drops, the need for a thinner oil at start-up increases, but the need for a thicker oil still depends on the stress and operating temperatures to which the engine is subjected. Hence multi-grade oils are perfect.
In other words the first number is for start-up at the coldest temperature, the second number is for running at the hottest temperature. i.e. your choice,
0W-20 (way way cold at start and engine never really gets fully warmed up)
5W-30 (cold start and regular operating temp, such as early spring or late fall)
10W-40 (cool temp start-up and regular temp operating - pretty much the normal range)
20W-50 (really hot days when engine might even overheat under normal operating conditions)
Multi-grade oil thickens only as temprature rises. Clutch rattle, camshaft rattle, and all manner of sound are dampened less by thin oil than thick oil. So, expect to hear more rattle in cold temperatures when using multi-grade oil.
