Both, depending on what part of it actually goes bad.
For me, it's almost always been a matter of the starter cranking slower and slower each startup, then finally the battery is just toast after sitting overnight. I'll replace the battery, and it'll be dead the next time I go to start the car.
If you're replacing it, I recommend doing the 3g upgrade if you haven't already.
I've had them go a number of different ways. Either you have a dead battery and go through what sgarlic said or a dummy light comes on or you start hearing funny nioses, like bearings squeeling. That was my latest. I had heard an intermitent niose but couldn't pinpoint it. Then an electric trans temp gauge sending unit went out for no reason. Turns out the alternator surged and killed the sending unit. It was just a matter if time before the alternator took a dump.