The BasicAnimator is for playing legacy animations. Essentially it allows you to build states that can be called from an Animate node or from code.
Many of these options mirror what is present in Unity already and are covered in depth on their Animation page.
The Basic Animator was an early attempt to handle the lack of an animation state machine in Unity (and to expose this to RAIN's behavior trees). A much better option is to use the Mecanim Animator and Animator to create a full animation state machine.