When a ball on a rope swings in a circle, is there both centripetal force and tension force?

The centripetal force can be made up of any type of force, whether gravitational, friction or tension. The centripetal force is not a force type, it is just a net force that is always radial. So it is a sum of forces, no matter the type.

So yes, it is a tension force. It just acts as a centripetal force.

The centripetal force is the force required to keep the ball rotating in a circle, by providing it the necessary centripetal acceleration. In this case this force is the tension in the rope, but of course in general the centripetal acceleration can be provided by forces of a different nature.

You might find it easier to avoid the term centripetal force and just state that there is a force due to the string (tension) which is producing a centripetal acceleration.

There are times when two (or more) forces are acting on a body, eg on a banked track, and the net force on the body in a particular direction produces a centripetal acceleration.
In such cases using the term centripetal force can be misleading as it might be thought to imply that there is only one force causing the centripetal acceleration.