I have a 2005 civic LX. I replaced the starter thinking that was my problem with the car not starting. That worked for about 24 hours. Now, again the car is not turning over at all and the battery is fine. I think I do hear the fuel pump kick on when I turn the key.
I think it may be the starter relay but have no idea where it is. It is not listed in the fuse/relay box under the hood. I'd appreciate it if anyone thinks it could be the starter relay and if so, where it is located.

 

Is the green key light under the fuel/temp gauges coming on at all when the car won't start? It should come on for a few seconds when you try to start and blink 3 times when you remove the key.

 

You should jump the starter to make sure the starter is OK and able to get power from the battery. Put transmission in neutral (or park if automatic), set parking brake, leave key off. Go under the hood and unplug the small wire on the starter. Connect a test wire from the small terminal on the starter to battery (+). The engine should crank.

Assuming that test passes, on manual Civics the switch on the clutch pedal is a common cause of not cranking.

 

Yes, green key light comes on and then blinks as you mentioned. What can I eliminate off the list of potential problems by that working properly?

 

Yep, the immobilizer is a common problem to have in 7th gen's that's why I asked. I would try what mk378 said next. Do you have a 5 speed or auto?