The conversation spoken about in the question above happens at the very beginning of the play in the first act. Theseus conquered Hippolyta's people and is now about to take her as his wife. It isn't quite clear as to whether or not she is being forced into this marriage, but many inferences can be made in their brief conversation before they are interrupted by Egeus.
Theseus is the first to remark that their marriage is "Four happy days" away (I.i.2). Notice that he marks the days before the marriage as a happy time. At that time, the summer's solstice will take place along with the revolution of a new moon. The new moon can also be symbolic for a new time period or change of life's circumstances as well as the changing of a month or season. Theseus and Hippolyta's wedding will change both of their lives by creating a new life as one, thereby making the connection to the new moon relevant. Then Theseus says that even though those four days will be happy, they will seem long to him because he is anxious to marry his bride.
Hippolyta responds with a slightly different perspective because she says the four days will pass quickly into night. The image of night can mean that her new life will seem like darkness as opposed to her previous one. Next, she focuses on the aspect of night rather than day by saying that those four nights associated with the four days will pass quickly like a dream. Hippolyta may be referring to the fact that since people sleep at night, and dream, that the time seems even shorter to her than at any other time in her life. She then references the moon as well by seeming to hold onto the last remaining sliver of her former life like the moon's final showing of its "silver bow" (I.i.9). She then says that the new moon will shine on the night of their wedding and not the day of, which might possibly suggest that she is more nervous about the wedding night than she is of the wedding day.
Summarily, if Theseus views four days as too long, and Hippolyta sees four nights passing too quickly, the bride and groom must be looking forward to the marriage in a completely different light than each other. Usually the one who can't wait for an event wants it to happen because she or he is happy about the event. Likewise, the one who thinks the time before the wedding is passing too quickly does not want the event to occur.