Im a little fuzzy on the difference between the two. If you look at the attached picture, if that force stays with the object as it rotates, like a hand pushing in the same spot, would that force cause an angular acceleration?
Since the force exerts a torque about the object's center of mass, it will produce an angular acceleration about the center of mass. The force will also produce an acceleration of the object's center of mass.
Ok, lets keep things simple. Way more simple. Draw a line on the box. That line will rotate an angle theta as you apply the force. That angle will get bigger and biger as you keep on applying the force. The rate at which that angle gets bigger is the angluar velocity. Same deal for the angular acceleration.
NOW, each point on the body will have a DIFFERENT tangential velocity. It will depend on how far out it is-i.e. the radius. At the very center of rotation, the tangential velocity will be zero. As you move further out, it will increase. This is because it is proportional to radius.
Consider this. All points on the body are rigid. As it sweeps an angle, lets say theta, all the points rotate. But the points on the OUTTER most edge must sweep a BIGGER circle in the same amount of time, hence they must have a higher TANGENTIAL acceleration.
If that dot in the center of your diagram represents a fixed axis about which the object is free to rotate (no friction, of course), then it's a much simpler situation. (You didn't mention any fixed axis in your original post.)
In that case the force exerts a constant torque about the center, which produces a constant angular acceleration. The tangential acceleration of each point of the object depends on its distance from the axis: [itex]a_t = \alpha r[/itex].