Metaphor is a figure of speech used to make comparisons between two very different things. The purpose is to attribute the characteristics of one thing to the other by saying they are one and the same - thing A is thing B. A metaphor does not use support words such as "as" or "like" to make the comparison. For instance, if I say, "Your lips are roses," I certainly do not mean that literally. Your lips are lips, but I wish to compare their color and texture to the ones of a rose. That is why I say your lips - thing A - are roses - thing B.
That is precisely what Shakespeare does in "Juliet is the sun." Metaphorically, Juliet is being compared to the sun. As we know, the sun is a fundamental factor for our existence. It provides us with the heat and the brightness without which we wouldn't survive. Juliet is as important as the sun for Romeo, now that he is in love with her. Without her, he feels that wouldn't survive.
