Underhill was the part of Hobbiton that lay directly beneath Hobbiton Hill. Its most famous smial was within the Hill itself; Bag End. Obviously Underhill contained Bagshot Row.
As Sauron had already learned of Baggins through Gollum and using the name outside of the Shire would prove dangerous, Gandalf told Frodo Baggins to use the name "Mr. Underhill" after leaving home.
Other versions of the Legendarium
In a draft, Underhill was a separate village. In such a draft, Bilbo's will mentions "Underhill near Hobbiton".
