Diagnosing car trouble can be a tricky process. However, keeping a few basics in mind will help you track down and identify most common problems. Cars require three basic things to work; spark, air and fuel. Everything in an engine is built to bring these three things together in the proper proportion. If one thing is missing or out of proportion, your engine won't turn over.
If your car wasn't having any trouble before you replaced the battery, but now it won't turn over you will need to verify that your new battery is indeed good. While rare, it does happen that you get a bad battery. If your battery burned out prematurely, you might want to check the rest of your car's electrical system, like the alternator, and make sure it's working well.
Check all the connections in the car, starting with the spark plug wires, then the distributor and then the ignition coil. If all these check out, you might want to pull your spark plugs and make sure they're still good and not corroded. Check out your distributor cap and make sure the connections are all still good.
Make sure your alternator is still good. Get it tested at a local parts store or garage.
If the car wasn't running before you replaced the battery, make sure it's getting plenty of air. Check the air filter and the hoses leading to and from the filter for excess debris or blockage. Also check the throttle body and intake manifold. If there is no obvious blockage, make sure the intake valves open and close properly.
Without gasoline, no internal combustion engine will run. First make sure you have gas in the tank. Then check all the fuel lines and make sure they are in good condition and securely installed. Open the hood and check the fuel rail and the injectors to make sure they aren't leaking and all the fuel lines are properly attached. If the car is old, you may need to replace the filter. If all of that is working as intended, get the fuel pump checked out.
