A week after the full moon the moon looks like a semi circle again?

Answer 1

Essentially yes.

The moon takes roughly 4 weeks to orbit the earth. During this period we see only the parts of the moon that are lit relative to the moon's position to the sun and the earth, as seen from the earth.

During a new moon, the moon is between the earth and sun, so only the "back side" of the moon is lit and we can't see it at all (and BTW, if the moon is DIRECTLY between the earth and sun, we have a "solar eclipse". As you can guess, solar eclipses can ONLY occur during "new moons").

A week later, the moon has orbited earth so that the moon is "next to" the earth relative to the sun, and we see half of the lit surface of the moon. This is a "half moon".

A week later, the moon has orbited so that the sun is on one side of the earth and the moon is on the other. This is the "full moon" and we can see the entire side of the moon that's lit by the sun. And BTW...when the earth is DIRECTLY between the sun and the moon, we get a "lunar eclipse". And as you can guess again, a lunar eclipse can ONLY occur during a full moon.

And again a week later, the moon has orbited earth so that the moon is "next to" the earth relative to the sun, and we see half of the lit surface of the moon. This is another "half moon", but unlike the half moon 2 weeks ago, the moon is on the OTHER side of the earth, so we see the OTHER half of the lit side of the moon.

Finally, a week after that, the moon has orbited to a point between the earth and the sun again, and we're back to a "new moon" again.

And one last "BTW", this roughly 4 week cycle of the moon's phases is where we get the word "month" from.