Surnames are added titles that you can have after your character name. Surnames are mostly for Role-Playing purposes.
To get a surname, you must be level 15 or greater. Visit a Notary and, for a small fee, you will be able to select a surname for your character.
There is a Notary NPC available for monster players in the Ettenmoors.
If you have a surname, and want it removed, uncheck the 'Show Surname' option. This option can be accessed by going to the UI Settings section of the Options Panel. Do so by holding down CTRL + O and clicking on the UI Settings.
There is no way to completely remove a surname once you have chosen one. A surname can be overwritten, but not removed entirely.
If you already have a surname and would like a new one, just visit a Notary.