I haven't personally hit anything (that I know of), but I had an experience like WillRavel's... though I didn't wreck my car. There was a bunny or opossum, not sure which, crossing the road... I lived at the end of a dead-end gravel road, so there wasn't any traffic, but I swerved hard and nearly ended up in the ditch. That taught me to never do that again.
Other related events:
--When I was 7 or 8, I was in the car when my mom ran over a local cat or dog. I just remember the thump, and the fact that my mom did not stop.
--Last summer in Zambia, my boss was driving me and a few other people to the water pump for our daily water run (the back of the truck was full of big barrels). She had stopped to pick up someone else in the village, and in the meantime the local kitty had lain down in front of the back wheel (we didn't know, obviously). After 5 mins the boss drove away, and as I sat in the truck-bed looking back, I saw the cat writhing in the dirt... at first I thought it was rolling in the dust, but then I realized it looked ghastly and unnatural in its twisting motions. It had been completely run over and was in its death throes, as someone else said.
The children in the village just stared and stared, and my boss started crying once she realized what happened (she loved that cat). It was very disturbing to see the cat die in that manner. By the time we came back from our water run, they had already disposed of the carcass, but my boss couldn't forgive herself for a long time.
I hope I never end up running over any animals, though it's probably bound to happen. I hate the feeling of killing something that was alive and innocent just a few seconds before. I know, I'm a softy.