A 9V battery is a "true" battery - that is, it is a device compose of multiple cells; what you buy at the store, commonly called "batteries" - aren't - they are "cells"; AAA, AA, C and D "batteries" are actually AAA, AA, C, and D "cells" - you need to connect multiple of these cells together to make a "battery" (likely, the terminology comes from the military, I bet - ie, gun "batteries" composed of multiple guns in a single emplacement).
Ok - so a 9V is a true battery (so are 6V and 12V lantern batteries - also car batteries are true batteries, as well as gel-cell batteries - do note, though, that at one time, it was possible to buy lead-acid cells in C and D sizes; not sure if this is still possible today, though); what kind of cells are inside?
AAAA - that's right - quad A - slightly smaller versions of the AAA cell. There exist a ton of other battery types (ever wonder why there isn't a B cell? Well, there is - or used to be - look it up), many designated by letters (N cells are interesting; there's also a certain cell out there that was for smoke alarms that supplied 22.5 volts and wasn't much larger than than a AA cell), others designated by numbers (or a combo, as is common with button cells).
I'd tell you go to wikipedia - but for the next 24 hours it is down (in the USA) due to protest against SOPA/PIPA (look those up too, if you care about the internet - and if you are in the USA especially)...