In act 1, scene 3, what is the purpose of the nurse?

The nurse serves multiple purposes, perhaps most importantly comic relief. Romeo and Juliet charges forth as a tragedy, with many characters - some beloved - dying by the end. With the nurse, Shakespeare gives his audience a chance to breathe before the next serious scene. Shakespeare plays with language, the nurse saying "bigger; women grow by men," hinting at a possible pregnancy if Juliet indeed marries Paris as Lady Capulet proposes in this scene.
During the course of the scene, the nurse aids Lady Capulet's plan to get Juliet to marry Paris, as she speaks of him in only the superlative: "Nay, he's a flower; in faith, a very flower." Shakespeare's tragedy needs Paris to challenge Juliet's relationship with Romeo.
The nurse also serves as confidante to Juliet by the end of the scene, encouraging her to "Go, girl, seek happy nights to happy days." This close confidence continues as the nurse is soon one of the only people who knows about Romeo and Juliet's relationship. By establishing the importance of this relationship early in Act 1, Shakespeare can later utilize the relationship to make Juliet even more desperate - desperate enough to try the friar's dangerous plan and careen the tragedy into Act 5.

The Nurse in this play is a comedic character, but what she says often reveals things of importance about Juliet's life and her relationship with her family. In this scene, she is a foil to Juliet's mother; her fond stories about Juliet as a small child stand in contrast to the stiff diction of Lady Capulet, who discusses the prospect of marriage with her daughter as if it were a business proposition. The apparent lack of intimacy between Lady Capulet and Juliet is strikingly different from the relationship between Juliet and the Nurse.
In discussing the meaning of marriage, the Nurse's opinion is opposed to Lady Capulet's: Lady Capulet states that "by having [Paris]," Juliet shall be "making yourself no less," referring to the increase in social status Juliet will enjoy as Paris's wife. Bawdily, the Nurse retorts, "nay, bigger, women grow by men," referring to pregnancy. Her focus is very much upon the physical—note that she repeats several times the joke once made by her husband that the child Juliet would "fall backward when thou hast more wit." This is a reference to Juliet "falling backwards" in order to allow a husband to have sex with her. For the Nurse, marriage is tied to sex and pregnancy; for Lady Capulet, it has an entirely different meaning and relates to social status.
Both the women, however, discuss marriage without really inviting Juliet's input. While Lady Capulet claims that Juliet is no longer a child, the fact that she intends Juliet to marry Paris is clear from the use of such modal verbs as "shall," as if this is a foregone conclusion. Meanwhile, it is clear from the story the Nurse tells of Juliet's childhood that she has, in her own way, viewed Juliet as a vessel intended to be married to a man from her earliest days. This scene shows that the relationship Juliet has with her Nurse is very different from the one between her and her mother, and yet it also appears that neither of them truly appreciates, or wants to understand, Juliet's own views about the question of marriage.