CLASSIFICATION
(SOURCE: NCBI database)

KINGDOM: Fungi
  DIVISION: Basidiomycota
CLASS: Agaricomycetes
ORDERS: Several

TOOTH
FUNGI

Mushrooms with spines or "teeth" don't form a single natural grouping of closely related species. Tooth fungi are simply fungi whose spores are released by basidia arrayed on the surface of small teeth, instead of on gills, the sides of pores, or flattish surfaces. Below is a fine one:

Hericium erinaceus, the Pompom Mushroom or the Lion's Mane

That handsome being goes by several common names, including "Pompom Mushroom" and "Lion's Mane"; It's Hericium erinaceus, one of the best-known tooth fungi because not only does it provide good eating but also is widely sold as medicinal.

Lion's Mane contains "... β-glucan polysaccharides, which are responsible for anti-cancer, immuno-modulating, hypolipidemic, antioxidant and neuro-protective activities," a webpage at the National Library of Medicine says, just as a starter.

The species mainly grows in temperate beech and oak forests of North America and Eurasia, though it's sporadic elsewhere.