Hello all.
I was messing around with my BIOS/CMOS settings, and I discovered that, after playing with one particular setting, my laptop no longer recognized when the A.C. adapter was disconnected.
So, for all you Inspiron 1501 users (And potentially other Inspiron users):
Keep the Quick Boot option disabled! It doesn't really save you much time at all, and it somehow messes up the ACPI events that report the status of the AC adapter.