Middle English kerven (the initial -k- is from influence of Scandinavian forms), from Old English ceorfan (class III strong verb; past tense cearf, past participle corfen) "to cut," also "cut down, slay; cut out," from West Germanic *kerbanan (source also of Old Frisian kerva, Middle Dutch and Dutch kerven, German kerben "to cut, notch"), from PIE root *gerbh- "to scratch," making carve the English cognate of Greek graphein "to write," originally "to scratch" on clay tablets with a stylus.
Once extensively used and the general verb for "to cut;" most senses now have passed to cut (v.) and since 16c.
carve has been restricted to specialized senses such as "cut (solid material) into the representation of an object or a design" (late Old English); "cut (meat, etc.) into pieces or slices" (early 13c.); "produce by cutting" (mid-13c.); "decorate by carving" (late 14c.). Related: Carved; carving. The original strong conjugation has been abandoned, but archaic past-participle adjective carven lingers poetically.