What did Juliet mean when she said deny thy father and refuse thy name?

Juliet says, "Deny they father and refute thy name," because she is willing to not let their feuding families hinder her and Romeo's romance. In saying this, she means that if he will forget that he is a Montague, then she will cease to be a Capulet, and they will be able to pursue the true love for which they feel for each other. She goes on to say: "What's in a name? That which we call a rose

By any other name would smell as sweet." This means that no matter the name you have, you are still who you are as a person. So basically, in summary, Juliet says this to tell Romeo that she couldn't care less that she's a Capulet and he's a Montague, that she loves him for him, Romeo. Simply Romeo.