Other forms: beasties
A beastie is a small animal, especially a cute, wild one like a chipmunk, or a beloved, domesticated one like a pet. Someone might even refer to an insect, such as a ladybug or a cricket, as a beastie.
As you might've guessed, beastie comes from beast, another word for "animal." The -ie added to the end is a diminutive suffix — it's used to indicate something or someone small, cute, or dear. You see it at the end of kiddie, for example. It often has the form -y, as in froggy for a little frog, and it's often attached to a person's name, as in Jeanie or Billy. There it expresses affection even if the person is not small — or cute.