The moon is tidally locked with the Earth, which means the same side always faces the Earth. So, for example, when the moon is between the Earth and the sun, the far side (from the perspective of Earth) would be fully illuminated by the sun.

The “far side” of the moon refers to the hemisphere that can’t be seen from Earth.

The sun. The moon always shows the same side to Earth, so the far side has phases just like the Earth-facing side does. When we see a full moon from Earth, the far side is a "new moon"; when we see a new moon, the far side is full. And so on for the other lunar phases.

The other well publicized photo they did was of the dark side of the earth (it was night), so same idea(?)

The sun still. It's just that that side never points towards the earth, but it still gets sunlight. Same as how the side we see isn't fully lit except during a full moon.

The sun