The euphemism is "euthanize." It's the term shelters use, though usually it's not the correct word choice. Used most precisely, "euthanize" means to painlessly end the life of an animal who is suffering and will not recover from that suffering.
Some shelters do not end the life of dogs and cats painlessly. Some shelters still use gas. I've heard of at least one that uses direct injection into the heart, but without first giving the dog or cat an injection to put him out. And then there's the Chesterfield "shelter" in South Carolina, currently under investigation for shooting dogs, perhaps even using them for target practice.
And as for the question of "suffering," thousands and thousands of pets who are perfectly healthy, including puppies and kittens, are "euthanized." Some would argue that any animal in a shelter, even if physically healthy, is suffering emotionally, is depressed, lonely, afraid, and if that animal does not find a home, his or her suffering will be endless. And there are not homes for all of them.
True on both counts. They do suffer emotionally. There aren't homes for all of them. There isn't enough room in shelters to keep dogs and cats forever. There aren't enough rescues to save all the animals on the "urgent" lists.
But language matters. We use "euthanize" to sanitize an ugly event.
Plenty of rescue workers and animal welfare advocates use other words. They send out pleas on Facebook and Twitter: Young friendly Beagle mix to be destroyed tomorrow. Urgent: 10 week old kittens will be killed this Tuesday. And some post the word "Dead" in red text over the photos of shelter dogs and cats that didn't make it out.
Maybe if everyone, including shelters, used these words instead of "euthanize," fewer people would so easily give up their pets to a shelter. Maybe fewer people would buy a pet and more would adopt, and thus puppymills would breed fewer dogs. Maybe more people would spay and neuter instead of believing their pet needs to have just one litter to be "fulfilled."
This was two year old Bentley. He was killed at a Florida shelter the same day this entry was posted.