I'm so sorry, sweetie. Your cat had a good long life--18 is quite old for a cat--and I'm sure you and your parents did everything you could for her. She no doubt was sick for a long time, and the cancer caught up with her.
A few years ago, I had to put down my 18-year-old cat, whom I adopted when she was 3. A few months before she died, an ultrasound to determine the cause of her frequent UTIs turned up an abdominal mass that turned out to be cancer. I didn't have a biopsy done because no way was I going to put a cat her age through chemo!
Long story short, one day when she appeared to be in great distress, my boyfriend and I rushed her to the vet where the doctor suggested "humanely letting her go". I wasn't surprised; I knew on the way over that we wouldn't be bringing her back. And while they were examining her my boyfriend and I agreed that if they suggested putting her down, we would do so.
I had the opportunity to say goodbye to her, so I gave her a few pats and told her I loved her, but she was out of it--she obviously was sedated--and so I called in the vet. The vet told us what would happen, what might happen, and injected the solution. It took less than a minute and was very peaceful. Believe me, I was crying hysterically and saying, "I'm sorry, I'm sorry." I am probably not the first person to cry hysterically as she said goodbye to her beloved furbaby. I think it is absolutely normal to be crying when you do this!
I think I was a wreck for a few days. I tried going into work the next day, but I just couldn't bear it and left early. I pretty much rested as much as possible before I had to go back to my next workday. Yes, it's normal to be that upset; you've lost a member of your family.
One website that helped me was www.petloss.com. There are message boards and a chat room where you can talk with other people dealing with loss. It helped me a great deal.
I also adopted another cat a couple of weeks later, and she has brought me comfort. It's not for everyone, but it's nice having someone to come home to, even if she doesn't say much. I don't see it as replacing Miss Kitty but as giving another cat a chance.
Take care.