Santorini really has some of the most unique beaches in Greece. Most of them, if not all of them, are covered by black sand which originates from the relationship of the island with the volcano. The most popular beaches are Perivolos, Perisa and Kamari but this is mostly due to the many cafes, bars an restaurants there.
Red Beach
The Red Beach is as you can see is basically...red! A very unique beach indeed. Not directly accessible though. You will have to stop near the sea and then go up and around the hill that hides this unique beach.
Vlychada (some spell is Vlihada) is a beach that has is
somewhat remote from the rest of the busy areas of the island. It
has some palm trees and it is also the location of underwater tours with
a submarine. Certainly a nice place to visit if you want to be
away from too much noise.
Kamari is one of the nicest beaches of the island. The beach
itself is more stony compared to other beaches, but it has something
that you won't find elsewhere on santorini: a long pedestrian road full
or bars, clubs, restaurants and cafes. 
Perissa is actually very close to
Kamari location-wise. Bu your way is blocked by a rocky mountain and you
will have to drive around it to get to the beach. It's a
relatively long beach, again with bars, etc and provides you with more
space on the beach. Ideal for beach sports.