NOTE: You may also be interested in the section that focuses on special-needs adoptions.
PetFinder [NOTE: This site is amazing in its capacity to help people find the right cat, dog, bird, or other creature to adopt or to help people place an animal they can no longer care for. There are photos and descriptions of over 50,000 animals needing homes. You can search according to geographic location, breed, size, gender, and age. You can also, if you want, limit such searches to special needs animals.]
Best Friends Animal Sanctuary: the largest sanctuary for abused and abandoned cats and dogs and other animals in the U.S.; over 1,800 on any given day. An excellent adoption program, including adoptions of special needs cats, dogs, horses, bunnies, and birds.
Save-a-Pet: Thousands of dogs, cats, and other animals in a database, updated daily, that can be searched by breed, size, color, age, and location.
Feline Rescue: can search for purebred cats of each breed or other categories; worldwide cat rescue directory; searchable U.S. cat rescue organizations by state.
Hearts United for Animals: a national no-kill shelter, sanctuary, and animal welfare organization.
Breed Rescue Network: for specific dog breeds.
Lucky Parrot Sanctuary: a place that gives abused and unwanted parrots a permanent home.
Utopia Animal Rescue Ranch: for dogs, a project of Kinky Friedman