The surname is sometimes habitational, used by someone who lived in any of several places of the same name. Otherwise it is topographic, referring to someone who lived near a wood. In both cases the name is from England or Scotland, and from Middle English.
The common noun comes from the name of the manufacturer, after John T. Underwood, one of its founders.
Underwood (plural Underwoods)
Proper noun 
- A surname.