This noun is a Latin loanword ( sudarium, from sudor, “sweat”) denoting small cloths used by the Romans for wiping the hands and face ( Illustrated Bible Dictionary, “Handkerchief,” 2:608).
The word appears four times in the New Testament. In the Parable of the Pounds, one servant kept his pound in a “napkin” (Luke 19:20, KJV; NASB, “handkerchief”; NIV, “piece of cloth”). In Acts 19:12 the sweatcloths (“handkerchiefs,” KJV, NIV, NASB) Paul apparently tied around his head while making tents were taken to heal the sick (Bruce, New International Commentary on the New Testament, Acts, p.384). In John 11:44 and 20:7 such “cloths” were used, according to Jewish custom, to cover the faces of the corpses of Lazarus and Jesus.