The proper name, taken from their castle in Swabia, is Staufen.
When the last member of the Salian? dynasty, Henry V, died without an heir there was controversy about the succession. Frederick and Conrad, the two current male Staufens, were grandsons of Henry IV and nephews of Henry V. After the death of the intervening king, Lothair III, in 1137, Conrad became Conrad III of Germany, king and emperor.
The last Hohenstaufen was never crowned emperor. After a 20 year period the first Hapsburg was elected king and emperor.