For an animal we would normally say "its eyes are small." However, in certain situations where the gender of the animal is specified, we might use the appropriate gender...for example, "The mother elephant always protects her young" or "My grumpy old cat is always impatient for his dinner." "Their" is used as a possessive adjective for a plural subject, for example, "All four of the girls brought their backpacks." However, in some cases it is also used as a singular possessive adjective when the gender of the person is unspecified, and I think this might be what is causing you to confuse it for "its." for example, "Every citizen must do their duty" - there is only one citizen being talked about, but because a citizen can be either male or female, some people will use "their" as a gender-neutral possessive adjective. Note that we would never use "its" when talking about a person as it would be considered both strange and rude.