It was raining hard and a car a mile or so ahead of us hydroplaned. So everyone was suddenly slowing down and I hit the brakes and then crashed into the car in front of me (it was my friend's car in front of me). My car had gotten totaled and he old got minor injuries to his car. No one was injured. The cops had come, got all of the information, and a tow truck had to take my car away because of the damage to it made it not drivable. No one received a ticket. My family came to get me and my friend was able to drive home in his own car. When we got the accident report it said it was a "no-fault" accident. Their auto insurance keeps sending me mail and wanting me to pay for it but my mother doesn't want to since it's no-fault. She has called them and talked to some lady on the phone and she didn't understand why we had to pay either. We even sent a copy of the police report to their insurance company.
My mother had found out the day of the accident that the company she thought was car insurance wasn't and was for the car company and the car but wasn't insurance (We got insurance soon after we found this out). Again I didn't receive a ticket and this was only my first accident and I was 17 at the time. So I have no idea what to do.

There is a difference between criminal ( getting a ticket) and being at fault, and actually being at fault for civil issues.

You should have gotten a ticket honestly, it would have been for following too closely, since by law, if you can not stop, no matter what the conditions, you are going either too fast for conditions, or following too closely.

So yes, you are at fault for the damages, Next not sure what thought it was insurance, sorry but how do you confuse if you have insurance or don't ? Sorry I find that hard to believe.

But yes you owe them for the damages, if you don't, they may sue you, it is possible to also lose your license for not paying for damages that you are responsible for.