Right... the O'Dowd clan then.
Originally spelt O'Dubdha (O'Dooda) they were a clan which rose to prominence among the Uí Faichrach people whom lived in west Ireland in around the 10th century, becoming the kings over the areas of county Mayo and Sligo. They retained kingship until the 13th century, and became confined to a barony in Tireragh. In the 17th century the O'Dubdhda ruling family were ousted from their lands, probably by Oliver Cromwell, leaving behind 24 castles and 52 towns. Soon after O'Dubhda became anglicised to O'Dowd, Dowd or Doody, O'Dowd being the most common.
As for on my mother's side...they came from the Lake District before moving down to the Wirral sometime in the late 19th century. (My great grandfather, on my dads side, emigrated to Liverpool in the 1920's) They could have been anything from labourers to free holders, but i imagine they were probably descendant from colonists who moved into the Lake District during the 13th century. Which means they must have originally lived in the north east of England. Given the family name Elwood, it sounds Anglo-Saxon to me, but given their location in the north east, there probably is some Viking blood in the family too.