"large amphibious quadruped rodent of the genus Castor," Old English beofor, befer (earlier bebr), from Proto-Germanic *bebruz (source also of Old Saxon bibar, Old Norse bjorr, Middle Dutch and Dutch bever, Low German bever, Old High German bibar, German Biber), from PIE *bhebhrus, reduplication of root *bher- (2) "bright; brown" (source also of Lithuanian bebrus, Czech bobr, Welsh befer; see bear (n.) for the likely reason for this).
Formerly valued and hunted for its secretions (see castor) and for its fur, which was used in the manufacture of hats, so much so that beaver could mean "hat" from 1520s and continued so into 19c. even after they began to be made of silk or other material. They were hunted to extinction in Great Britain in the 16th century but have lately been reintroduced.
"lower face-guard of a helmet," early 15c., from Old French baviere, originally "child's bib," from bave "saliva."
"female genitals, especially with a display of pubic hair," by 1927, British slang, ultimately from beaver (n.1), perhaps transferred from earlier meaning "a bearded man" (1910), or directly from the appearance of split beaver pelts.