Use Petfinder.com's search tool to find the kind of pet you want, then contact the shelter listing that pet and follow their adoption procedures. You cannot adopt a pet directly from Petfinder.com because it only lists dogs for other shelters and rescue organizations.
Adoption procedures vary depending on the rescue group you work with, but many require a preliminary application and interview. During this process, the group asks questions about your lifestyle, any other pets you own or have owned, and what your home is like. They may verify this information by contacting your references or talking to your landlord if you rent your home. Some rescues require you to let a representative visit your home to make sure it is appropriate for the pet you want.
Once you have initial approval, you can meet the animals available for adoption. This generally includes the animal you saw on Petfinder.com, but sometimes that animal has already been adopted. This process lets you get familiar with different animals and find one you like. After you have made your choice, the rescue generally requires you to sign an adoption contract stating that you can care for the animal, pay an adoption fee and bring your new pet home. Most require you to bring the pet back to the rescue if you cannot keep it.