I'm learning about centripetal force and I was shown a scenario where a man was spinning a rope attached to a ball over his head. There's a centripetal acceleration toward the center and therefore a force must be acting in the same direction as the centripetal acceleration — the centripetal force. But I wonder if that's considered a tension force as well. Is it?
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.
The tension in the rope is what provides the centripetal force.
Tension exists simply because the rope or string that you're whirling around is extended.
The adjective "centripetal" describes the direction of the force. Centripetal means "center seeking". The origin of the force may be gravitational (earth - sun system) or electromagnetic (e.g. nucleus attracting electrons) but the direction has to be towards the center.
Firstly get clear what is centripetal force about. It means nothing, it is just the resultant of all forces during circular motion and this resultant force is known as centripetal force.
Now, you are saying that tension and centripetal force is same in your case or not, the answer is very simple that because the resultant vectors of all forces in your scenario are in direction of tension force thus the centripetal force seems to be in direction of tension force.
You need to remember that there is nothing called a separate "centripetal force". Any force can be the centripetal force in a circular motion. So, in your case, the centripetal force is provided by the tension in the string. When a biker goes around a Loop of Death in a circus, the normal reaction is what gives the centripetal acceleration.
So, when you say, "But I wonder if that's considered a tension force as well. Is it?", it is more so the case that the tension force is considered the centripetal force here, and not the other way around.
Hope this helps!