Battery light indicator on in the dashboard

on my 2004 Honda CR-V with about 132K miles, the battery light comes on and stays on for some time. Then goes away. It was ignored for some time. Yesterday the car lights, dash went out, but CR-V reached home. I checked the battery (changed in Apr/08), it had no water. I added distilled water. The CR-V did start. I tried twice. This morning, the headlights come on. But CR-V does not start! is it alternator or battery or both. It is worth to charge the battery overnight and check what happens? Any other SIMPLE tests?

No water in the battery? This usually means the battery is being overcharged and the electrolite has been boiled away.

Recharge the battery. Start the engine and with voltmeter, test the alternator output voltage at the battery posts. If the voltage reads over 15.5 VDC, the voltage regulator has failed. Unfortunately, the voltage is regulated by the Power Control Module which means the engine computer in your vehicle will require replacement for the overcharging problem.

Tester

Simple tests?! The simplest tests I can think of are with a multimeter.
Charge, or replace, the battery. Run the engine at 1500 rpm and check the charging voltage. If it’s over 15 volts, the battery is being over-charged.
There is a slim chance that you may find out why it’s over-charging by checking the wires between the alternator and the engine computer (ECM). Check with your handy dandy multimeter for resistance, in the wires, from the alternator to the ECM. Resistance shoudn’t be more than a couple of ohms. Check the wires which connect to the ECM terminals B10, B13, and B18.
After these simple tests, you may decide to turn it over to a competent shop for repair.