Middle English sinne, from Old English synn, syn "violation of divine law, offense against God; moral wrongdoing," also "injury, mischief; enmity, feud; guilt, crime, misdeed," from Proto-Germanic *sundiō "sin" (source also of Old Saxon sundia, Old Frisian sende, Middle Dutch sonde, Dutch zonde, German Sünde "sin, transgression, trespass, offense," extended forms).
The notion is probably ultimately "it is true," i.e. "the sin is real" (compare Gothic sonjis, Old Norse sannr "true"), from PIE *snt-ya-, a collective form from *es-ont- "becoming," present participle of the root *es- "to be."
The semantic development would be via the notion of "to be truly the one (who is guilty)," as in Old Norse phrase verð sannr at "be found guilty of," and the use of the phrase "it is being" in Hittite confessional formula. The same process probably yielded the Latin word sons (genitive sontis) "guilty, criminal" from present participle of sum, esse "to be, that which is." Some etymologists believe the Germanic word was an early borrowing directly from the Latin genitive. Also see sooth.
The details of the purely theological definition are much contested. Sin-eater is attested from 1680s, "one who, for pay, takes on the sins of a deceased person," typically by eating certain food in the presence of the corpse. To live in sin "cohabit without marriage" is from 1838; the phrase was used since Middle English in a more general sense (to sin with has been "commit fornication or adultery with" since c. 1200). Ice hockey slang sin bin "penalty box" is attested from 1950.
