Current without Voltage and Voltage without Current?

What flows is not the voltage but the charge, and that flow is called current. There can be voltage without a current; for instance if you have a single charge, that charge induces a voltage in space, even if it's empty. Voltage, in the most physical way, is a scalar field that determines the potential energy per unit charge at every point in space.

Now, you can't have currents without voltages because if there's a current there's a charge moving, and every charge produces a voltage, but you can have currents without voltage differences in space. For example, if you have a charged sphere, and you make it rotate, the charge will be on the surface and by rotating the sphere you will have a current on the surface, but the voltage is the same at every point of the surface. Also magnetization of materials can induce currents by the same way.

If you introduce a charge in a circuit without a voltage it just doesn't move?

That's true, it won't move, unless you have some changing magnetic field that may introduce "voltage differences" between the same point, making $\nabla\times E\not =0$, although that wouldn't be electrostatic voltage the way you're seeing it.

When you think electricity think water. Let's use a waterfall as an example for this analogy:

The Water traveling from the high point to the low point of the waterfall is like electrons flow through a conductor. That is the current: current == flow.

Voltage by definition would is the "difference of the potenticals" in the waterfall analogy the voltage will be between the highest point and the lower point of the waterfall. The higher the raise the higher the voltage. When one point is more charged with electric than the other, that's voltage.

If the waterfall is dry and thee is no Current, the difference between two points is still there. One point is higher than the other (one point is more electrically charged than the other). That's voltage without current.

If voltage was zero (if the high point and low point of the waterfall were on the same level) would water still fall down? No, water would stay still. Still = no flow = no current without voltage.

Hope this helps :)