What Is a core-core type transformer?

A 'core type' transformer core is one in which the primary windings and secondary windings are placed around each of the limbs, as opposed to a 'shell type' core, in which the two windings are placed across the centre core. A core-type core is a magnetic circuit equivalent to an electric series circuit, whereas a shell-type core is equivalent to an electric parallel circuit. This is better answered with a diagram, so I suggest that you do a search on the internet.