This button is in the new Firefox, but what is the name of it?
This type of button is usually called a 'Split Button'.
The glossary of the Windows User Experience Interaction Guidelines defines the term 'split button' as:
A bipartite command button that includes a small button with a downward pointing triangle on the rightmost portion of the main button. Users click the triangle to display variations of a command in a drop-down menu. See also: command button. [ http://www.microsoft.com/downloads/en/details.aspx?displaylang=en&FamilyID=e49820cb-954d-45ae-9cb3-1b9e8ea7fe8c ]
MSDN's Ribbon Framework Control Library defines a 'Split Button' as:
The Split Button is a composite control with which the user can select a default value bound to a primary button, or select from a list of mutually exclusive values displayed in a drop-down list bound to a secondary button. [ http://msdn.microsoft.com/en-us/library/dd940505(VS.85).aspx ]
Jquery UI calls it a 'Split Button' - http://jqueryui.com/demos/button/#splitbutton
Not a bad name, I would say.
They've also been called Muttons - https://web.archive.org/web/20121116015828/http://hanamidesign.com/blog/ux-pattern-mutton-menu-button/
Sorry, I don't know the formal term for this type of button but in my team we tend to refer to it as a 'multi(call-to)-action button' ... I am certain there is a better name than that however!! It is used extensively in the Apple store also, so as a consequence I suspect will be seeing them ever increasingly in the future! ;)