What a great question!
We have a rotating space station shaped like a wheel with some radius r, rotating with a constant angular velocity ω = R, and two astronauts. We'll say the astronauts, named Fred and Jorge, have the same mass. Fred is standing at some computer terminal, and Jorge is running laps so he can look trim for all the handsome people on Earth. Neither of them is accelerating. Let's talk Newton's laws. His second law states that the sum of the forces acting on any object is equal to the mass of that object times its acceleration (F = m*a). Acceleration is the change in velocity with respect to time.

Before we go any further, let's define our directions, or coordinates. Let's say the astronauts can move in two directions, either towards or away from the center of the space station or tangent to the circular space station. Tangent means perpendicular to the line connecting the center of the circle to any point on the edge.

Now let's think about the sum of the forces action in these directions. Newton informed us that the sum of those forces on any object will equal the mass of that object times its acceleration. Are the astronauts accelerating tangentially to the center of the space station? Well, the space station is rotating at a constant angular velocity, and the tangential velocity is equal to the angular acceleration times the radius of the space station (v_tan = ω*r). Since the angular velocity isn't changing, the tangential velocity isn't changing, and so the their is no tangential acceleration. Likewise, Jorge is running at a decent clip such that he isn't moving with any angular velocity. As long as he runs at a constant velocity, his angular velocity doesn't change, and so his angular acceleration is zero as well. The total force acting on Fred and Jorge in the tangential direction is consequently zero.

For something to change the direction in which it's moving, it has to accelerate. So if there is no net force in the tangential direction, there must be a net force acting in the centripetal direction (at least in Fred's case).

What do things look like in the centripetal (center-pointing direction)? In the world of things moving in a circle at a constant speed, it happens that the acceleration in the centripetal direction is equal to the square of the tangential velocity divided by the radius of the circle (a_cen = v_tan²/r). Since v_tan = ω*r,a_cen = ω²*r. Fred is moving at a constant angular velocity R. The net force acting on Fred in the centripetal direction is equal to Fred's mass times his angular velocity squared times his distance from the center of the space ship (F_net,cen,Fred = m_Fred*ω²*r). Jorge isn't moving with an angular velocity, so the net force acting on him in the centripetal direction is equal to zero.

Centripetal sounds a lot like centrifugal, doesn't it? That's no coincidence. While centripetal means center pointing or toward center, centripetal means away from center. Centripetal force is a real force, centrifugal is fictional. To rotate around in a circle at a constant speed, you have to accelerate towards the center. That acceleration is known as centripetal acceleration. The feeling you have of being flung away is the centrifugal force, which is just a consequence of you being accelerated toward the center.

The two astronauts do not feel the same centripetal or centrifugal force. Fred is accelerating centripetally, and Jorge isn't. And since they are getting their feeling of weight from the centripetal force, Fred feels heavier than Jorge, even though they have the same mass. For more information about the difference between the centrifugal and centripetal forces, check out this website:

hyperphysics.

I am picturing the astronaut doing something similar to this move clip: watch this video, except he should be running pretty fast instead of walking. Let's call this Olympian sprinter Jerry. If Jerry did indeed match the angular rotation with his sprinting, he would no longer experience centrifugal force.

What would happen to Jerry then? Since he is essentially floating, the force of his feet pushing on the space station as he tries to run would send him towards the middle of the space station. If he had nothing to grab on to or slow him down, he'd float right on to the other side. It's doubtful he'd reach the other side feet first, so instead he'd bump into the spinning space station at some odd angle. Since the space station is moving pretty fast, he's in for a nasty rug burn."