There is no single answer as to the composition of a basketball in that there is no single kind of basketball.
The inner lining of basketballs tends to be a rubber or synthetic rubber bladder that maintains the air pressure within the ball.
The external covering of the ball varies depending on the intended use of the ball.
A leather external cover is used when precision of handling is required. The leather gives a true bounce and grips the floor better when dribbling and passing which allows for better control with spin passes or direction changes. This covering is most commonly found in professional use.
Synthetic leather is more durable and can be used outside on rough courts. It isn't quite as reliable in bounce and response but is less expensive and lasts longer.
Rubber exteriors are used on low end basketballs that one might find on a school playground. Durability is their main draw.