"excavation in earth for reception of a dead body," Old English græf "grave; ditch, trench; cave," from Proto-Germanic *grafa-/graba- (source also of Old Saxon graf, Old Frisian gref, Old High German grab "grave, tomb;" Old Norse gröf "cave," Gothic graba "ditch"), cognate with Old Church Slavonic grobu "grave, tomb," and perhaps from a PIE root *ghrebh- (2) "to dig, to scratch, to scrape," related to Old English grafan "to dig" (see grave (v.)). Or perhaps a substratum word in Germanic and Balto-Slavic.
The normal mod. representation of OE. græf would be graff; the ME. disyllable grave, from which the standard mod. form descends, was prob. due to the especially frequent occurrence of the word in the dat. (locative) case. [OED]
From Middle Ages to 17c., they were temporary, crudely marked repositories from which the bones were removed to ossuaries after some years and the grave used for a fresh burial. "Perpetual graves" became common from c. 1650. Grave-side (n.) is from 1744. Grave-robber attested from 1757. To make (someone) turn in his grave "behave in some way that would have offended the dead person" is first recorded 1888.
1540s, "influential, respected; marked by weighty dignity," from French grave (Old French greve "terrible, dreadful," 14c.), from Latin gravis, "heavy, ponderous, burdensome, loaded; pregnant;" of matters, "weighty, important;" of sounds, "deep, low, bass;" figuratively "oppressive, hard to bear, troublesome, grievous," from PIE root *gwere- (1) "heavy."
In English, the sense "solemn, sober" is from 1580s; of immaterial things, "important, serious" 1590s. Greek barys (opposed to kouphos) also was used figuratively, of suffering, sorrow, sobbing, and could mean "oppressive, burdensome, grave, dignified, impressive." The noun meaning "accent mark over a vowel" is c. 1600, from French.
"to engrave," Old English grafan "to dig, dig up; engrave, carve, chisel" (medial -f- pronounced as "v" in Old English; past tense grof, past participle grafen), from Proto-Germanic *grabanan (source also of Old Norse grafa "to dig; engrave; inquire into," Old Frisian greva, Dutch graven "to dig, delve," Old High German graban, German graben, Gothic graban "to dig, carve"), from the same source as grave (n.). Its Middle English strong past participle, graven, is the only part still active, the rest of the word supplanted by its derivative, engrave.