A rare earth is an oxide of a rare earth element. Often rare earth elements themselves are loosely called "rare earths".
As to exactly what is a "rare earth" element is a matter of some ambiguity. The narrowest definition restricts it to the lanthanides, which are lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, and lutetium. Sometimes the radioactive actinides are included, so that all elements in the f-block are considered "rare earth metals" in a manner similar to how d-block elements are called "transition metals". The actinides are actinium, thorium, protactinium, uranium, neptunium, plutonium, americium, curium, berkelium, californium, einsteinium, fermium, mendelevium, nobelium, lawrencium. All are radioactive, and only thorium and uranium are found in large amounts in nature because of their very slow decay rate; one might argue that the elements from neptunium on cannot be properly called "rare earths" because they are all artificial. Lastly, some include scandium and yttrium as rare earth metals because of their chemical similarity to the lanthanides.
The cheapest rare earth metals are yttrium, lanthanum, cerium, praseodymium, neodymium, samarium, and gadolinium, along with thorium and uranium; while the most expensive are scandium, europium, thulium, and lutetium.