There are numerous ways to get your new family member including a rescue, breeder, or pet shop. We highly recommend choosing your pet from a rescue or a breeder.
Animal shelters and rescue agencies, or what used to be known as pounds, are either governmental or private organizations that provide temporary homes for stray, surrendered, or abandoned pet animals. The animal is kept at the shelter until it is reclaimed by the owner, adopted to a new owner, placed with another organization, or euthanized.
Unfortunately, resources are seldom adequate to support the large number of animals taken in by these organizations. As a result, animals that are not claimed by their owners, or that have temperament or health issues that cannot be corrected or treated within the resources of the organization, are often euthanized. Shelters that receive a disproportionate number of animals compared to available adopters may also euthanize animals because of space concerns.
A small number of shelters have chosen to be "no-kill" shelters, which support healthy and adoptable pets for the remainder of their lives or until they are adopted. However, as funding is limited, the number of animals that can be accepted by these organizations can be low, and some animals may not be accepted because of behavior or health concerns. "No-kill" shelters often do euthanize if they receive animals with these problems. There are no clear standards for assessing these issues, and so statistics cited about how many "adoptable" animals are euthanized or adopted can be meaningless.
Animal control agencies, or nonprofit organizations contracting for animal control duties, also enforce animal-related ordinances. Some animal shelters also provide low-cost spay/neuter surgeries or veterinary care, behavior training, or other services.
By contrast, animal sanctuaries will look after animals for the rest of their natural life, without necessarily attempting to find them any other home. Some establishments combine the qualities of an animal shelter with those of a sanctuary.
A great resource to find animals available for adoption is Petfinder.com.