Mozilla started with Mozilla Application Suite, then moved on to create Firefox. Mozilla was abandond, but picked up again as SeaMonkey.
Firefox was mainly much safer than IE for a long time due to the lack of market share. However, as that changes more and more exploits which target firefox are being found in the wild.
Firefox has a rather fucking slow JavaScript core. It also relies heavily on JavaScript for the UI. (It is, after all, a XUL based application, using JS for all the UI) This creates more UI hangs than IE, Safari, and Opera suffer from.
I use firefox out of lack of a more favoured option for me.