Vaulted describes a ceiling or roof that rises in a continuous arch, creating a sense of height and openness inside a room. Arched is broader — it can describe any curved shape (an arched doorway, an arched bridge, arched eyebrows) and does not always imply a ceiling. Domed specifically means shaped like a half-sphere or dome, as in a domed stadium or a domed cap; a vaulted ceiling is typically a series of arches rather than a single hemisphere. Curved is the most general term — it applies to any line or surface that bends, from a curved road to a curved smile, and lacks the architectural weight of vaulted. In short: vaulted signals an arched ceiling or roof with architectural presence; domed signals a hemisphere; arched signals any arch shape; curved signals any bend.
From the past participle of the verb vault (to build an arched roof), which came into Middle English from Old French vouter. The Old French verb derives from voute (a vault or arch), which goes back to Latin voluta, the feminine form of volutus (rolled or turned), from the verb volvere (to roll or turn). The architectural sense has been the dominant meaning since the 14th century.