This is one of those issues like religion and politics that can create heated exchanges. The goal of most shelters and rescues is to find good homes for homeless animals. There are shelters/rescues for pigs, birds, horses, reptiles and every other domesticated and pet animals but I am going to focus mainly on the dog and cat shelters.
In this link you will find the differences between No-Kill, Low-Kill and Kill shelters:
Here is a parable about no-kill policies: You’re walking next to a river and you see a kitten floating past. You jump in and save the kitten. Then another one floats by, so you save that one, too. Then another and another and another float by, and you soon realize that you can’t save them all. So you run upstream to see who’s throwing kittens into the water—and you stop that person.
The sad reality is that there are approximately 600 dogs, puppies, cats and kittens killed each hour of the day in the U.S. Because there is simply not enough room in the shelters for them. Yes, this is absolutely unacceptable but is there a solution or a way around it? Hence the debate begins!
Some shelters euthanize just to make room for the new animals coming in. Some don’t euthanize at all. It may sound all fuzzy and warm to never euthanize an animal but what about those that are terminally ill? What about the animals that are so aggressive that they are beyond adoption? Who is to blame? Did you spay/neuter your cat or dog? Did you train them and do you really provide them with a good home forever? Are you part of the problem?

Here is a very informative and interesting article I came across while researching:

Answer to the Difficult Euthanasia Question
September 25, 2003
Every now and then I receive a letter asking why the LA/SPCA is still euthanizing animals and when are we going “no-kill.” Many people are afraid to ask the question, so I thought I’d share the substance of my response with you.
Every city has the unfortunate task of collecting stray and unwanted animals. The LA/SPCA assumes that duty for New Orleans and receives up to 1,000 animals each month. Shelters like these are called “open admission” shelters. They accept animals regardless of injuries, pedigree, or reason for surrender; they provide refuge for all. “Limited admission” shelters are often called “no-kill” shelters because they do not euthanize. These shelters cannot accept any and all animals as they would be forced to euthanize because of the sheer number of animals that require housing. Consequently, limited admission shelters must ration their intake, which leaves hundreds more animals for another agency to accommodate.
Unfortunately the inflow of unwanted animals continues to be far greater than the number of available homes for those animals. What is the most humane way to address the needs of all these surplus creatures? Shelters across the country accept 6-8 million cats and dogs each year. If the LA/SPCA alone receives a thousand animals in a month, where would they stack the animals until new homes are found? Sadly, there are not enough homes and there is not enough sanctuary land to house 6-8 million pets each year. Imagine if the nation stopped euthanizing for 4 years. Over that period, the country would be housing 24 million homeless dogs and cats. Is that reasonable? Is warehousing in the best interest of the dogs and cats? This is the difficult debate caused by owners who have neglected or refused to have their pets sterilized.
The LA/SPCA believes that every animal deserves a high quality of life. Living in cages over the long term does not support this principle. Since an agency in New Orleans must take in these animals, the LA/SPCA accepts the responsibility. If an animal must be euthanized, the LA/SPCA compassionately puts them down via injection, a traumatic method for staff but the most sensitive and dignified for the animal.
Euthanasia and sheltering are not the solution, but a temporary necessity. Spaying, neutering and education are the only answers to the deep rooted problem of overpopulation which is why the LA/SPCA invests so heavily in sterilization programs. Until all dogs and cats are sterilized or the numbers of homeless animals are significantly reduced, our community will continue to euthanize.
The ultimate goal is to be a city where adoptable animals are no longer euthanized by any agency. Until then, someone will be required to perform the tough task.
– Laura Maloney

C.C.H.S. Is a very small shelter. We are not no-kill but instead are low-kill. Very low kill to be precise. We can only provide for a small amount of rescued animals. We rely heavily on foster parents and volunteers and have ZERO paid employees. The more fosters we have the more dogs and cats we can help. We also rely on donations and fund raising. You won’t believe how much volunteering just a couple hours a month helps. Yes, this is an informative article but it is also a plea for help. Finding home for needy dogs and cats is gratifying but it’s kinda like bailing out a sinking boat with a spoon. The boat will sink unless you fill the gaping hole in the bottom. Finding a home for a cat or dog saves a life but spaying/neutering saves hundreds if not thousands of lives by preventing generations of homeless/unwanted animals. Spay and neuter clinics help. Spay and neuter laws help. It’s like the hole in the boat or the person throwing the kittens in the river. Stop it at it’s source and the problem greatly diminishes and possibly ends the need for euthanization. If you have questions or would like to volunteer please give C.C.H.S. A call @ 618-526-4500 or stop by 1301 Apple Lane in Breese. Before I go I will add a shameless plug. We have quite a bite of merchandise at C.C.H.S. We have shirts (for humans), collars, leashes, toys, treats and much more. Most of which is on sale. Shameless plug over. Thanks for reading. Until next time, enjoy.