Its definately both. Your alternator died, and it wouldnt go down alone. it grabbed the battery, and took it out too.
You see, when an alternator fails, it no longer charges the battery. So, the car was running off the battery alone....and now it is a gonner.
Your battery starts the car...and keeps it running. The alternator, with built-in voltage regulator, re-charges the battery at about the same rate that the engine draws from it....add in other electrical draws(stereo, heater, lights, computer, etc), and the alternator has to step up, and charge more.
well, what happened was, you alternator said "screw this...I give up"
your battery said "whoa man...I cant do this by myself", so... stereo-off, climate control-off, and it barely made it home
you shut it off, and the battery said "thats it. I quit...darned alternator abandoned me..I aint doing this by myself... I quit".
so, go get both. The old ones are gone.
alternator...dont waste your time with a reman...get a new one from Ford.
battery-make sure you get one with enough CCA for the car(cold cranking amps).
Good luck, and be glad it died at home...you got lucky.