Precious metals include gold, silver, platinum and palladium (primarily traded on the CME Group's exchanges) - whereas base metals are things like copper, aluminum, zinc, nickel, etc. (primarily traded on the London Metal Exchange - LME). Base metals by definition are metals that rust or corrode more easily than precious metals.
As the name suggests, precious metals are more expensive than base metals. Base metals are used mostly in industrial production. Other examples of base metals are cobalt, titanium, tin, aluminum, etc.