Origins
The word 'chin' is one of the oldest words in the English language, traceable with high confidence to the Proto-Indo-European period. It descends from Old English 'cin' or 'cinn' (chin), from Proto-Germanic *kinnuz, from PIE *ǵenu- (chin, jaw, cheekbone). The PIE form is reconstructed with great confidence because of the regular cognates across nearly every branch of the family: Latin 'gena' (cheek), Greek 'génus' (γένυς, jaw, chin), Sanskrit 'hánu' (हनु, jaw), Avestan 'zanu-' (jaw), Old Irish 'gin' (mouth), Gothic 'kinnus' (cheek), and Tocharian A 'śanwem' (jaws).
An intriguing possibility is that PIE *ǵenu- (chin, jaw) and PIE *ǵónu (knee) are related, both deriving from a more abstract root meaning 'angle' or 'joint' or 'bend.' The chin is the angle of the jawbone; the knee is the angle of the leg. This connection is supported by the near-identical form of the two roots and by the semantic parallel — both denote hinge-like joints in the body. If the connection is valid, then English 'chin,' 'knee,' 'genuflect,' and 'gene' (from Greek 'génos,' perhaps from the same root complex) would all be distantly related.
Notice the semantic drift across the IE branches: where English uses *ǵenu- for the 'chin' (the point of the lower jaw), Latin 'gena' means 'cheek,' Greek 'génus' means 'jaw' more broadly, and Gothic 'kinnus' means 'cheek.' The same PIE root was applied to different parts of the lower face in different daughter languages — chin, jaw, or cheek depending on the tradition. This kind of semantic wandering within a defined anatomical zone is typical of body-part vocabulary across language families.
Old English Period
The Old English 'cin' underwent the regular palatalization of /k/ before front vowels that is characteristic of English — the same sound change that turned 'cild' into 'child,' 'cēosan' into 'choose,' and 'cyrice' into 'church.' This is why English has 'chin' with /tʃ/ while German preserves the original /k/ in 'Kinn.'
The 'chin-up' (an exercise in which the chin is pulled above a bar) dates from the 1950s in its exercise sense, though the phrase 'chin up' as encouragement ('keep your chin up' — maintain a brave, upright posture) dates from the 1930s. 'Chinstrap' is self-explanatory. The informal 'chinwag' (a chat) is British slang from the early twentieth century, from the image of the chin wagging (moving up and down) during conversation.
Anatomically, the human chin — the bony protrusion at the base of the mandible — is unique among primates. No other primate, and no other hominin species (including Neanderthals), has a true chin. The reason for this distinctively human feature remains debated: proposed explanations include structural reinforcement for the forces of chewing, sexual selection, and a byproduct of the face becoming flatter as the human jaw reduced in size over evolutionary time.