apostrophe (n.1)
"mark indicating an omitted letter," 1580s, from Middle French apostrophe, from Late Latin apostrophus, from Greek apostrophos (prosoidia) "(the accent of) turning away," thus, a mark showing where a letter has been omitted, from apostrephein "avert, turn away," from apo "off, away from" (see apo-) + strephein "to turn" (from PIE root *streb(h)- "to wind, turn").

In English, the mark often represents loss of -e- in -es, possessive ending. By 18c. it was being extended to all possessives, whether they ever had an -e- or not.
apostrophe (n.2)
"a turning aside of an orator in the course of a speech to address briefly some individual," 1530s, from Middle French apostrophe, from Late Latin apostrophus, from Greek apostrophos, literally "turning away," from apostrephein "avert, turn away," from apo "off, away from" (see apo-) + strephein "to turn" (from PIE root *streb(h)- "to wind, turn"). Related: Apostrophic; apostrophize.