No, the shape of the moon stays fixed. It is only the appearance of the moon (as seen from the earth) that changes every night. This perceived change in its shape is due to its rotation around the earth and the fact that we only see the portion of the moon that reflects the sun's light towards the earth.
The moon does not have any light of its own and reflects the sunlight hitting it. Our observation of the moon (and hence its shape) is dependent upon the amount of sunlight reflected by the moon (towards the earth), which in itself is a function of the relative positions of the moon, the earth and the sun. When the earth is between the moon and the sun, the moon is able to reflect light from its entire lit portion, thereby giving us a full moon. At the other end of this cycle, when the moon is between the earth and the sun, we hardly see any portion of the moon and hence observe a new moon.
Thus, the moon does not change its shape every night, it is only the portion that reflects the light towards the earth.
Hope this helps.
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