So this is the underside or oral surface of Spike, a purple urchin (Arbacia punctulata) that I had in a tank long ago. See his pretty purple-tipped spines?
He was attached to the glass with his tube feet (those long tentacle-y looking things with round suction tips). The larger, striped tube feet with slightly indented suctions that surround the mouth are the buccal podia used for "tasting" stuff before chomping on it. The green fleshy part around the white pointy bits are essentially Spike's "lips" or peristomial membrane (peri = around, stoma = mouth).
The five soft, pinkish lobes in the center of the mouth are called paradental tongues (thank you Drs. Iain Wilkie, Michael Russell, and Joe Pawlik for helping me identify these structures!).
The aforementioned "white pointy bits" are the five white "teeth" that Spike was using to scrape algae from the glass. These teeth are the protruding tips of the Aristotle's lantern (the rest of the lantern is internal). The teeth and tongues work together to manipulate the food into pellet form before ingesting (Bonasoro & Carnevali 1994- thanks again, Dr. Wilkie!). Mmmmm- pellets.
According to Grzimek's Animal Life Encyclopedia (Volume 3: Mollusks and Echinoderms; 1972), the Aristotle's lantern was so-named by Pliny the Elder (A.D. 23-79). I bet Pliny was probably beat up by his Roman classmates since he was clearly a nerd and his full name was Gaius Plinius Secundus (bless his heart and thanks Wikipedia). But I guess all those guys had pretty silly names back then so maybe Pliny was never subject to whatever the Roman version of a "swirly" or "wedgie" entailed.
Anyway, here is a great illustration (by Diller?) from the Grzimek text showing all the little "bones" of the Aristotle's lantern.