I Connected The Positive And Negative Battery Cables Backwards And Then Started The Car. Now It Will Not Start. What Could I Have Damaged And How Do I Fix It?

Okay, first thing I would ask myself is; what receives power from the battery?
Let's see, that would be the starter, starter relay switch.
Okay, would suggest you check the fuses first.
Hopefully it has only blown a fuse, or starter relay switch.
Maybe, it only blew a fuse or relay before it did any damage to a major component such as, the starter.
Don't know anything about what type of vehicle you have, so this is the best I can give you.

Well, I'm surprised you got as far as you did without major sparks. The cars of today are so sensitive to current because of all the electronics, computors etc.. All of these things are fuse protected except for the starter, ignition coil, solenoid switch. Check all fuses. Unfortunately, its going to be a process of elimination. Good luck

I did the same to my ford ranger and now the lights come on but the truck wont turn over.. Radios dead and I replaced some fuses that blow out.. Any ideas as well

The car should of not started you need to check an see if you didnt kill the battery by going straight to ground