Husserl's transcendental consciousness is what reamins after the phenomenological reduction and Epoche. This means that after doing away with everything that is not experience based, we are left with a remainder or a "residue" in Husserl's terms). This thing that cannot be reduced or suspended, but which isn't a phenomenon in a regular sense, is a form of pure or transcendental consciousness.
The reduction aims to conquer “transcendental consciousness” or pure consciousness, that is to say, a consciousness which constitutes within itself the transcendence which characterizes reality itself .