This quote is spoken by Juliet in Act Three, Scene Two. As Juliet waits in the orchard for her Romeo to come to her, she is instead met by her Nurse, who is there to deliver some horrific news: Juliet's cousin, Tybalt, has been killed by Romeo in a duel, and Romeo is being banished from Verona as a result of this violation of the Prince's decree against violence in the streets.
The particular lines you have mentioned (and the long soliloquy that precedes them) occur at the very opening of the scene, before the Nurse has spoken her grave message. In this passage, Juliet is speaking of her deep love for her new husband and how she longs for him to meet up with her so that they may consummate their marriage. Juliet approaches the end of this speech, claiming:
...And when he shall die,
Take him and cut him out in little stars,
And he will make the face of heaven so fine
That all the world will be in love with night
And pay no worship to the garish sun.
The literal interpretation of these lines would be a plea to the gods to turn Romeo into a constellation in the sky upon his death so that she (and the rest of the world) may always admire his incredible beauty. Metaphorically, Juliet is asserting her commitment to the boy and her total (perhaps naive) admiration of him. At the same time, in referencing Romeo's death, these lines seem to foreshadow what is to come later in the play: Romeo will die, but the nature of this death and the legacy of love that he leaves behind will have a startling impact on Verona, inspiring the Montagues and Capulets to finally seek peace.
Juliet is explaining how physically attractive she finds Romeo. She is saying that if his body was the stars in the sky, people would love the night time as much as she loves Romeo. In general, day time would have been considered better than night time. Juliet is saying this would change if the stars were as appealing as Romeo. Of course, Juliet is speaking of Romeo's physical attributes because she doesn't really know him at this point in the play. Keep in mind that Juliet is about 14 years old at this point. As a teenager, her mind is easily infatuated with an attractive boy.
Professor Levenson(Oxford ed.) wrote: "Gibbons notes the parallel between Romeo's celestial transformation and Julius Caesar's, the last of Ovid's METAMORPHOSES." The rest is science.