Check all under-dash and under-hood fuses with a test light or multimeter:
http://www.askmehelpdesk.com/cars-tr...tml#post252145
Pay particular attention to:
. Power distribution fuse (80A). It may be under-hood Fuse No. 41.
. ACG (ALT) (S) fuse (15A). It may be under-dash Fuse No. 24.
I suspect the brushes are worn, rotor slip rings need cleaning, or the voltage regulator may be bad. You need to replace or rebuild the alternator. Here's how to replace the alternator:
http://www.askmehelpdesk.com/cars-tr...tml#post224856
If you are adventuresome, here's how one common Civic alternator can be rebuilt:
http://www.askmehelpdesk.com/cars-tr...tml#post666293
I enjoy rebuilding alternators--it's cheaper and more satisfying. Every do-it-yourselfer should do it at least once.
My gut is that the brushes ($3.50) are so worn that the alternator can't charge the battery. I've seen this on alternators and generators. Replacing the brushes is frequently all that's needed. Afterwards, the Charging System Light goes out and everything works fine.