You need a new battery. The alternator needs the battery to create the
magnetic field it uses to generate electricity. With a wheezy battery
your voltage will be low, so you'll get low-reading gauges and a poor-
to non-running engine. A really crapped-out battery can also act as a
"lowest common denominator", dragging the voltage down even when
jumpered to another running vehicle.

> A battery that I've had a bad history with wen't dead on me---no surpise there.  The alternator is good, I'm pretty sure---it's new, and last time this battery wen't dead, I tested it, and it checked out.  I think the battery is a dud.  (probably from sitting around for a couple of months)
...
> Then, of course, it won't start, since my battery is dead, but it won't take a jump either---I tried for a long time a number of different times, and the starter motor was turning over quite quickly.  I waited 30 minutes (for a tow truck) and tried again... no luck.