In Act 1, Scene 3, Juliet and the Nurse are presented as having a very strong bond, even a friendship and mother-daughter relationship. However, it's also very clear that Juliet finds aspects of Nurse's personality irritating.
We especially see the closeness of their bond from the Nurse's perspective in this scene. We learn from one of Nurse's speeches that she lost her own daughter who was the same age as Juliet, which enabled Nurse to become Juliet's wet nurse. Since Nurse took care of Juliet rather than her own daughter, Juliet became like a surrogate daughter to her, as we see in Nurse's lines:
Susan and she (God rest all Christian souls!)
Were of an age. Well, Susan is with God;
She was too good for me. (I.iii.22-24)
The fact that Nurse thinks of Juliet in the same context of her late daughter Susan shows how attached she has become to Juliet. In addition, her exact knowledge of Juliet's age as well as her rambling story about weaning Juliet also show how fond of Juliet she is. Hence, we can see from this one speech that Nurse feels very motherly towards Juliet, which shows us a lot about their relationship.
Juliet, on the other hand, does not say a great deal in this scene that portrays her feelings for or bond with Nurse. However, we do see her feeling embarrassed by her Nurse's weaning story, just like a maturing child often feels embarrassed by his/her parents. We see her embarrassment when Juliet begs Nurse to stop her story. Juliet's embarrassment helps us see their mother-daughter-like bond. In addition, we know that only Nurse raised her as a child and not her own mother. So naturally, Juliet would feel a stronger bond with Nurse than with her own mother. Hence we see from Nurse's speeches and Juliet's reactions to what her Nurse says, that they have a very strong bond and friendship, a lot like a mother-daughter bond.
The Nurse is an interesting character whom I often relate to as the "comic relief" character, especially in this scene. The Nurse played the role of Juliet's primary caregiver, and was also Juliet's wet nurse (breast fed her). We know The Nurse has some signifigant attatchment to Juliet, as she basically raised her, but you can't help but read her part as almost a "best friend" type character. She willingly, and freely talks about Juliet losing her virginity, and repeatedly laughs and jokes about the story of Juliet falling down and hitting her head (which leads to the sexual intercourse talk). We have Juliet who is very weary of making committment and getting married, and The Nurse suports this decision for the most part. The Nurse, though, is loud, obnoxious, and very inappropriate. It would not be too far off to say that The Nurse is Juliet's driving force to make a rash and impulsive decision to give Romeo a chance and eventually marry him. In Scene 3 we get more of who The Nurse is as a character, rather than the relationship her and Juliet share, but we can assume that they share one that is close and free.