Double check the connections at the battery. Be sure and clean the connections with a stiff wire brush and apply a light coat of grease when you reconnect the battery posts.
--Modified
If that doesn't work you need to check your ignition switch!!!
Try checking the gorund cable at the opposite end from the battery. I had one come loose on my sporty and it would not start but the lights would come on.
Check the battery cables or starter.
You probably have a bad ground. Check the grounds on the brake lights.
I had this problem on my 1995.. it turned out to be the crank sensor I would check the battery terminal, did you clean it good when you replaced the battery? Also I have seen defective batteries and to check it put it on a load tester for batteries. I think most parts stores carry them and will check it for free. I would start with that battery terminal first if you haven't done that.
When you car is turned off your battery doesn't have the alternator pushing that constant stream of voltage to your battery to keep it charged. So your looking at a decrease in voltage by 2 - 3 volts while car is off. No part will fix this just simply start your car before you turn on your head lights.
Weak battery? Loose or corroded battery cables? Starter solenoid?
Loose or corroded battery cables Weak battery
Needs a new battery.
Is anything else going on? Will it start? If not take the battery cables off and reattach them. This is a common problem that I have had to address with mine. With mine nothing electrical would work as well, no lights, radio, etc... If I shook the cable the lights would click on and then it would die again when I tried to start it until I removed the cable and replaced it.
no, usually it will be a fuse. a battery would simply dim the lights to beyond usefulness, and if it gets too low, the pcm may start killing unneeded components.
First of all is the lights working in the car? (over head lights or any kind of lights) If your not getting anything at all (no lights) then you should check the battery. If the lights are working then I would check your starter. Most of the time if you take the battery somewhere to get checked they will do it for free. So I would start there. Did you try to get a jump to see if that would start it? Hope this helped some.
Sounds like a bad alternator. The alternator runs the accessories as well as the charging of the battery, whereas the battery is resposible for start up. If you had to charge your battery that also means that alternator wasn't doing it's job and needs to be replaced. Its not a big deal to swap it out.