Profiles are a way for people to share information about themselves, what they do, the things the like or dislike, special places they recommend, and so on. Profile information (as is the case with most things) is stored on Second Life servers.
In Firestorm, profiles my be displayed in one of two ways:
The look and feel of these is very different so they are treated separately; click the links above to access the section you are interested in.
To select which of the two you wish to use, access Preferences → User Interface -> Interface Windows (the Preferences window can by opened with Ctrl-P, or from the top menu bar, Avatar → Preferences). Then enable Use web profiles by default to use web based profiles; disable it to use legacy profiles.
See this page for documentation on Firestorm 4.6.9 (42969) and earlier.