There are 17 elements that make up the rare earth elements: scandium, yttrium, and elements 57-71. Elements 57-71 are also known as the lanthanide series or just the lanthanides, so lanthanide series is probably what you are looking for. Those elements are lanthanum (for which the series is named), cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium and lutetium. Rare earth elements are all metals so they are sometimes called the rare earth metals.
Rare earth metals aren't really rare in abundance compared to some other elements such as gold. They probably got their name because they occur in low concentration in deposits and are difficult to mine and separate.
Rare earth elements are used in rechargeable batteries and a number of consumer electronics products. The demand for many of these elements increased with the advent of color television. They're also used by the military in defense equipment.