I notice most of the focus in the thread so far has been on human life, but I'd like to add that, while I feel animal euthanasia is acceptable in some cases, I think people overuse it. I know many people use it when they don't want to be burdened with an aging, expensive, or inconvenient pet and they feel rehoming would be too difficult.
I have a dog that I was asked repetitively "why don't you put that thing down?" She has genetic leg deformities in her hind legs that cause her to look and move a bit off. We did look at surgical options at one point, but it would have taken about a 5th of our yearly income, and the success rate wasn't all that great. It wasn't an option. We ended up getting her another dog to play with to stretch those muscles(she was becoming too sedentary), and that worked great. She moves around like a three legged dog. She is not in pain, and manages just fine on the stairs(we have a lot of them here).
If a person would put her down, it would be because her condition bothers them, not her.
Another example I can remember was a cat we had years back. He was elderly, and had a myriad of health problems. One morning we woke up, and it was clear the cat would not last the day. We thought it would be a matter of hours. We talked about having him euthanized, but my husband had a gut feeling this wasn't the right thing to do. He was my oldest son's best friend in the whole world, and we thought maybe he was hanging on to say good bye(my son was at school when we discovered him). Well, my son got home, and they spent some time together. We left to get dinner, thinking that the cat would have departed by the time we got home. He hadn't. My son set up a "going away party" the cat. They played bored games, and read books, and had snacks(the cat couldn't eat them, of course). Eventually my son fell asleep. Within a few minutes of my son falling asleep, the cat passed on. In my view, it was clear that cat had wanted to hold out for his friend, and spend some time together before he shuffled off. To have euthanized him that morning would have interfered with his rights and wishes.