Is normal reaction for an object at rest, the component of frictional force perpendicular to the surface of the object? I know that for an object at rest there's no frictional force acting on it and that frictional force arises because of another force trying to displace it (Static friction). Gravitational force is not a contact force then, 
My textbook says;

the component of contact force normal to the surfaces in contact is called normal reaction. The component parallel to the surfaces in contact is called friction.

The component of which force is normal to the surface of the body? If the body is at rest there can't be any frictional force.
My second question, isn't friction an opposing force, then why is it mentioned as being a parallel component of a force in contact (friction itself is a contact force),if the contact force isn't friction what's the force they are referring to?
 A: 
I know that for an object at rest there's no frictional force acting on it

That's not correct.  An object at rest may be under the influence of a static friction force.  
Imagine a brick sitting on a ramp.  It is at rest.  Why does it not slide down the ramp?  

if the contact force isn't friction what's the force they are
  referring to?

In the ramp example, because the brick is not accelerating, we can assume that all forces sum to zero.  Gravity provides one force and the only other thing touching the brick is the ramp.  So the ramp must be exactly cancelling out gravity.   
Don't get too hung up with the fact the quote is talking about a general case (where parallel and normal components might both be present) and a specific case (such as a horizontal table).  In the horizontal case, we can still say that friction is the parallel component, just that it will be zero.
The important thing is that contact force equals normal component plus parallel component.  It doesn't say that they're all non-zero.

This force from the ramp is the contact force.  Now, it is often convenient to divide this force from the ramp into two separate components: a normal component and a parallel component.  

Here, the "friction" and "normal" forces are just orthogonal components of the ramp "contact" force.

If a body is at rest and no force is applied on it then it means that
  it has 0 force of friction.

If it has no other forces, then you're correct.  However in this example, the block does have another force on it, gravity.  Both the normal and friction components act to counteract the gravitational force.
A: The textbook states that there is a general contact force, made of two separate forces or components. One of them is normal to the surface, and the other is parallel.
Friction and normal forces can be considered contact forces alone, but some textbooks prefer to treat their vectorial sum as a single general contact force.
