The nursery rhyme is mainly a plot device. The original rhyme "Ten Little Indians" concerns the unpleasant fate of a group of ten young Native Americans. At the start of the nursery rhyme, there are ten of them, but gradually, one by one, they come to a sticky end, until there's only one left— and he hangs himself.
Agatha Christie's use of the nursery rhyme foreshadows what happens in the story. Ten guests have been invited to a remote island, and in due course, each one ends up dead. Indeed, many of them end up suffering almost identical fates to the poor, unfortunate boys in the nursery rhyme. For example, in "Ten Little Indians" one of the boys manages to chop himself in half, leaving only six left. In "And Then There Were None" Thomas Rogers, the butler, is found dead after going outside to chop wood.