Throughout Midsummer Night's Dream, the moon and moonlight occur frequently, clearly making them a governing symbol of the play. It can be said to broadly symbolise love, lust and fantasy, though interestingly it is used differently by different characters throughout the play. However, for all of them, the moon and moonlight is linked to bizarre and out-of-character behaviour - it seems to encourage weird behaviour.
In Act I Scene 1, Theseus laments the amount of time that has to pass before his union to Hippolyta, saying:
... four hapy days bring in
Another moon: but oh, methinks, how slow
This old Moon wanes; she lingers my desires
Like to a step-dame, or a dowager,
Long withering out a young man's revenue.
The moon here then is compared to a step-dame or a dowager who is represented as a block to the union (we presume sexually as well) and true love of Theseus and Hippolyta.
In response to Theseus' impatience, Hippolyta replies:
Four days will quickly steep themselves in night:
Four nights will quickly dream away the time:
And then the Moon, like to a silver bow,
New bent in heaven, shall behold the night
Of our solemnities.
It is clear that Hippolyta is not necessarily in such a hurry. She regards the moon as a witness to their pleasures of their marriage, and of course the sexual union between them. Interestingly, the moon was considered to be a powerful force of fertility.
One of Egeus' complaints against Lysander is that he has wooed his daughter by serenading her at moonlight. To Egeus, then, the moon is associated with foolhardy and reckless acts of love - he links moonlight with Lysander having "bewitched" Hermia, as if he is blaming the malign influence of the moon for Hermia's defiance of his wishes.
Lastly, Lysander tells Helena of his and Hermia's plans to elope, he says:
Tomorrow night, when Phoebe doth behold
Her silver visage, in the watery glass,
Decking with liquid pearl, the bladed grass
A time that lovers' flights doth still conceal...
For him, then, he clearly regards the moon as a force that supports the cause of true love by aiding him and Hermia in their escape. The moon's visage seems almost to bless the elopement of the lovers.
Throughout the play the majority of the action happens in moonlight or it is referred to as a force of love or chaos, specifically later on in the play. It is perhaps significant that after the "Midsumer Night's Dream" the marriages of the three sets of lovers occur in daytime, perhaps reflecting the end of the power of the moon to disrupt and cause chaos.