I'm going to attempt to cover domestic and wild animals as well as present lethal, non-lethal and preventative alternatives for dealing with them. This is by no means a definative or exhaustive list, these are just the methods that I've used or heard about being used sucessfully. I make no guarentees as to their efficacy in your specfic situation.
That being said let the lecture begin...
Step 1: Domestic Animals
The best way to get rid of a pet you don't want is to take it to your local animal shelter. There are many non-profit groups that run shelters for animals you can no longer care for. By simply opening the phone book or doing an internet search you can easily find one. They don't charge to take the pet and are usually very sympathetic. Some are "no-kill" shelters, meaning they won't euthanize the pet if it isn't adopted within a certain period of time.
If you have a pure-bred dog or cat variety another option is breed specfic rescues. These are groups that focus on one breed of cat or dog. If there is a breed there is a rescue for it. These groups will take the animals, and work to find them good homes. My wife and I work with a greyhound rescue and find it to be very rewarding. (Check out the Grey Hound Connection to find a group near you.)
I don't know much about large animals (e.g. horses) but I'm assuming rescues that deal with them exist as well. I'm sure a little research can turn one up. My brother had the good idea of checking with 4-H or Future Farmers of America for large animal rescues in your area.
Like most things in life prevention is the best way to avoid the problem. In order to avoid having to make a painful choice like this there are a couple of things you can do:
1) Don't impulse buy. I'm guilty of it myself. The damn creature is just so cute you buy it without thinking or researching. This is how you end up with an animal that isn't right for you or your lifestyle. Also be sure to look at your finances ahead of time. To properly care for a pet is costly. I heard somewhere that for each pet you have your annual expenses increase by $1000 (U.S.). So you need to decide if you can afford another mouth to feed, vaccinate, and provide vet care for before you buy.
2) Have your pet sterilized. Dogs and cats give birth to multiple offspring at a time. Multiply that by the above figure and ask yourself if it is really too much trouble to have your pets spayed or neutered. It also relieves you of having to make the painful choice of which adorable fluff-ball to give to the shelter. And many shelters will spay or neuter pets at reduced rates.