When I was about 9 years old and living in Scotland, we got two mischevious cats in our house, one tabby called Daisy (my sister's cat) and one ginger rascal called Jasper (my cat). Alas, Daisy died at a young age but Jasper lived to build quite the reputation in our household and we had a lot of adventures with him. He went for walks with us and even followed us to school once and started to climb up the fire escape looking for me (I was mortified at the time and thought about pretending he wasn't mine but then felt very proud of him and cheered him on until our teacher noticed and put a stop to that).
One day, when Jasper was about 6 years old, he disappeared for a couple of days, and I eventually found him curled up in a grass nest looking pretty sad. My mom knew something was very wrong and having a fair bit of experience with these things (being an animal nurse at one point in her career), told me that it was likely that Jasper was off to the vets for the last time since she suspected he had cancer. I couldn't bring myself to go with her and waved goodbye to Jasper bawling my eyes out (I am tearing up now remembering it) but I knew that it was the right thing for Jasper. It was my first real pet death at the age of 14 and a very difficult thing to accept.
I tell you this story to illustrate how each of us at some point will be faced with making some tough decisions on our pet's health and life. Deciding to put your pet down and going through with it is one of the hardest things anyone has to go through in life. I came across a brochure on pet euthanasia by the American Veterinary Medical Association, which might be helpful in these emotional decisions.
Your veterinarian understands human attachment to pets, and can examine and evaluate your pet's condition, estimate its chances for recovery, and discuss its potential disabilities and long-term problems. He or she can explain medical and surgical options and possible outcomes. Because your veterinarian cannot make the euthanasia decision for you, it is important that you fully understand your pet's condition. If there is any part of the diagnosis or the implications for your pet's future that you don't understand, ask to have it explained again. Rarely will the situation require an immediate decision and usually you will have some time to review the facts before making one.
In addition, here's another article that explains more about the actual process of euthanasia.
"The typical euthanasia is an overdose of anesthesia that suppresses brain activity and stops
the heart so that the animal doesn't feel any pain. Once the heart and breathing stop, death
follows shortly. It generally happens in less than a minute," explains Dr. Eurell. "It is possible
that there might be a physiologic reaction that can cause some concern for an owner
observing the death. The pet might go into an excitatory phase where there might be
muscular movement. Also, the sphincters may relax and the pet might urinate or defecate."
In addition, animals may not close their eyes at the time of death. These activities are not
indications that your pet has suffered pain.