David Slater is from Glouchester, so U.K. copyright laws apply to him. The monkey is from Indonesia, so Indonesian copyright laws apply to him.
A group of monkeys apparently were playing with Slater's cameras when one monkey started clicking photos. Slater did not assist the monkey in creating the photos and, according to Slater, most of the hundred or so photos that were taken by the monkey were blurry.
Here are the relevant Copyright Acts:
For the U.K., where "author" is defined as "the
person who creates" the work:
http://www.legislation.gov.uk/ukpag/1988/48/contents
For Indonesia, again with
person used in defining the author of a work:
http://www.wipo.int/wipolex/en/profile.jsp?code=ID
Under no copyright law are animals able to claim copyright in a work, although it could potentially be possible for a human to get copyright protection for a work they
helped an animal to create. Slater, however, already cut that possibility off from consideration by his earlier accounting of how the monkey took the photo.