Is ISS orbiting with or against the rotation of Earth? Is it important in any aspect for the space station?
The ISS orbit is prograde - in the direction of the Earth rotation. Prograde orbits are orbits with inclination less than 90 degrees. ISS inclination is 51.6 degrees.
These orbits are slightly easier to reach, because they don't require as much fuel, as you get additional "kick" from the Earth during launch. Retrograde (opposite direction) orbits are rare, because they require more fuel to reach. It would be inefficient to build the ISS in retrograde orbit, because all the craft would have to spend additional fuel to get there. The ISS orbit was chosen to make it accessible from the key launch sites in United States (Florida) and Kazakhstan (Baikonur).
It orbits same direction as Earth. You can expect all space stations always to be build to orbit in direction of earth. Saves fuel to reach them since you get extra boost while leaving Earth and less full again when slowing down when reaching the station. If it would rotate against earth the speed would be twice as night relative to you and you would need to spend more power to slow down. Or if you would chase it lose the extra boost leaving earth and waist there no fuel.
Antzi, in the screen shot at https://i.stack.imgur.com/30Eau.png says the Space Station's orbit follows the Earth's rotation around the Sun. Rotation is around an internal axis. Revolution is around an external object. The questioner wondered if the space satellite revolved around the Earth in the same direction as the Earth rotated around its axis. The answer is yes, and the orbit is inclined 51.6 degrees.