Sometimes (but not always) you can determine, given the condition of the animal, whether s/he's lost or abandoned:

If the animal is groomed, appears well-fed and well-socialized , especially after a thunderstorm or fireworks, he is likely lost. In this case consult all of the same resources a person who has lost a pet might utilize. See our My Pet Is Lost section.
Screen carefully in your search for his/her family. Dog-fighting rings use cats and dogs as bait animals, testing labs pay for donated animals, and so on. For example, omit identifying characteristics from postings or require that anyone claiming the animal first bring photos or past veterinary bills as proof.
An astonishing number of animals are left behind when their families move, or are abandoned in rural areas. Many of these animals are hit by cars, starve to death, or end up as prey for coyotes. These animals were born into a dependency on humans and whether you volunteered for it or not, you are now this animal's greatest hope for survival.