What do people actually mean by "rolling without slipping"?
You can always decompose a motion like this into two parts: (1) rolling without slipping and (2) slipping without rolling.
What is slipping without rolling? It means the object moves uniformly in one direction along the surface, with no angular velocity about the object's own center of mass. For instance, a box that is pushed along the ground can easily slip without rolling.
Unfortunately, most people seem to assume that you can infer some physically important information from your own notion of what slipping is, without having to define it. I believe this is done to try to connect to intuition, but in the process, things get a lot more nebulous and ill-defined.
To me, it's easier to think about this in terms of the object's rotation--it was never obvious to me that the point in contact with the ground doesn't have velocity at the instant it touches. I prefer to think instead that an object that rolls without slipping travels 1 circumference along the ground for every for every full rotation it makes. And object that travels more than this distance (or that doesn't rotate at all) is slipping in some way.
Then, eventually, we can get to the notion that the point in contact during rolling cannot have nonzero velocity through whatever logical or physical arguments necessary.
But as is usual in physics, it's not really clear what definition should be considered "fundamental" with other results stemming from it. This emphasizes that physics is not built axiomatically.
The above answers are all good but i want to give another example which really helped me with understanding what does it mean that the point of contact has a velocity of zero.
Think of the 'spinning circular object' not as a ball, but instead think of it as a star polygon with an infinite amount of edges, for the sake of the example a very large number will suffice - 9 edges
At any given moment only one of the edges is touching the ground. Think of the star's movement - if it's not slipping, the point that is touching the ground is not moving, it pushes against the ground, 'fighting' the force of friction.
Another nice example is the human cartwheel, but it has 2 points touching the ground at the same time, which is why i like it less...
This is why wheels are good for transporting things, as a result of the contact point not moving only the static friction force is working on the wheel, and static friction maintains conservation of energy.
I hope this gives another point of view to anyone googling the subject and running into this question...
If the wheel is rolling without slipping, what's the velocity of the point at the base of the wheel?? It is... zero! Convince yourself that the velocity must be zero. Since if it wasn't zero, the wheel wouldn't be rolling without slipping.
So far the explanation is correct. "No slipping" refers really to some non-zero interval of time, and to the state of the contact faces during this time. When there is no slipping the faces can exert greater tangential force on each other than in the state of sliding, and there is no loss of mechanical energy involved.
This can happen when two bodies are in physical contact for some non-zero time and the parts in contact have the same velocities during that interval.
It is not good to define no slipping just for an instant via the requirement that $v' = 0$ at that instant, because that can happen even if the bodies slide on each another at all other instants.
So the wheel is rolling without slipping if and only if the point at the base has velocity zero,
the word "so" is not very good here, and it should be added at the end that "contact point has velocity zero _all_the_time". Then it is OK.
It is interesting though that in practice there seems to be no case of perfect no-slip rolling. There is always some slipping and hence friction involved, even the train wheels on rails do slip a bit. The no-slip condition is thus a convenient approximation.