In circular motion, it is said that the friction acts towards the centre of the circular path. But friction opposes tendency of relative motion and since the velocity of the body(in circular motion) is along the tangent to the circular path at that instant, why isn't the frictional force acting in the opposite direction.
In the absence of any other centripetal forces on an object on a circular trajectory, in order to stay on its circular path a friction force pointing towards the centre of the circle provides the centripetal force $F_c$.
This friction force does indeed oppose (and if sufficiently high, prevent) motion of the object in the radial direction. Think about a car navigating a circular bend, staying on track.
This does not exclude other, tangential, friction forces being in play, like air drag, rolling resistance etc.
You may already know that in uniform circular motion, a centripetal force must exit to keep the car on its circular path. Like Gert mentioned that force would be a frictional force and in case of a car it is between the tires and the road. You should realize that a centripetal force is simply a property that must be assigned to one of the forces acting on the car. If you were to only examine a two dimensional view of the tires and the road's contact surface you would see that the weight, the reaction force and the friction force are the only forces acting there. The friction force is the only one in the direction of the centripetal acceleration and hence it is the centripetal force.
Moreover, the velocity you mention is in fact the tangential velocity which is constant (hence uniform circular motion), thus no tangential acceleration exists and no force in that direction to be opposed by a frictional force.
The velocity that does change is the angular velocity which results in the centripetal acceleration and hence the centripetal force.