The cube falls into the category of volumetric displays, meaning that it can be used to represent 3 dimensional shapes. It is composed of 8*8*8 512 RGB LEDs, namely the very popular WS2812 that you can find in Adafruit’s neopixel product line.
It is sold by a company called Looking glass factory. It is still a product at the kickstarter stage. You can read about the story of its development on this instructable.
They make an 8*8*8 and a 16*16*16 version. The small version will set you back 399$, not something that I can afford. I was able to play for a while with the one from WearHacks. The solution for you can be even more rewarding: build your own! You can start by having a look at these Instructables.