Wood is mostly a place surname: Middle English wode meant woodland, so “atte wode” families became Wood. It points to living in or beside the trees, the same topographic habit that gave us Hill, Brook, and Field.
A rarer English nickname from a different Old English word meaning “frenzied” or “wild” is noted in dictionaries, but the woodland reading is the main one. Woods with a final -s is often the same idea. Immigrant names meaning “wood” in other languages (French Bois, Polish Las/Les, and so on) sometimes became Wood in the US. Lead with the forest-edge story.