What is inside of a conductor?

A conductor is defined by the property that the electrons are fairly mobile so they are able to redistribute themselves in order to cancel the electric field due to any excess charge. But the electrons are bounded by the surface of the conductor or the work function. So you can still have accumulation of charge on the surface that creates an electric field outside the conductor. So inside can be thought of as the region in which the electrons are mobile.

Conductors are mostly solids which have free electrons in it which allows conductivity through it. So inside conductor there are atoms which have free electrons. Depending upon the no. Of free electrons we can tell which conductors has more conductivity.

A common misconception is, that physics has an official precise definition of each and every term that is used. Physics is not mathematics. If you want to understand what you read, you have to read carefully what the respective author writes. Sometimes it is written explicitly, most often it is only written between the lines. Language is imperfect outside mathematics. So what is meant by "inside" and "outside" in the literature you are using, can only be answered by the author of the text, or you as the attentive reader.