In Romeo and Juliet, what does Juliet mean when she says "That which we call a rose / By any other word would smell as sweet"?

In this instance, Juliet is referring to the importance of a name in contrast to the essence of the object to which it refers. She is essentially saying that even if a rose was called something else it would retain its characteristics such as, for example, its sweet smell. She emphasizes that the name of something does not change its nature.

Juliet makes this remark in Act 2, scene 2. She has learnt that Romeo,...

In this instance, Juliet is referring to the importance of a name in contrast to the essence of the object to which it refers. She is essentially saying that even if a rose was called something else it would retain its characteristics such as, for example, its sweet smell. She emphasizes that the name of something does not change its nature.

Juliet makes this remark in Act 2, scene 2. She has learnt that Romeo, the young man she is infatuated with, is a Montague. She is a Capulet and the two families have been involved in a squabble with each other for generations. This means that Romeo is her enemy and that she should not, therefore, have any association with him.

Juliet is quite distressed about the situation but does not allow it to affect her feelings for Romeo. Her sentiment is an expression of how deeply she feels for the young man. She has been overwhelmed by him and does not care whether he is a Montague or not. As far as she is concerned, it is not Romeo's name that makes him who he is. He could have had an entirely different name and she would have loved him all the same.

She has previously stated, though, that she would prefer it if he should abandon his name and deny his heritage. If he cannot do so, she will be happy if he should declare his love for her. She will then, in turn, forsake her own name:

O Romeo, Romeo! wherefore art thou Romeo?
Deny thy father and refuse thy name;
Or, if thou wilt not, be but sworn my love,
And I'll no longer be a Capulet. 

She does not believe that being a Montague means that she should dismiss him. Montague, is after all, only a meaningless name and does not constitute any integral part of a body. Romeo shares this sentiment. He expresses his own sentiments after hearing what Juliet has said about the matter:

I take thee at thy word:
Call me but love, and I'll be new baptized;
Henceforth I never will be Romeo.

The two impetuous lovers then embark on a very dangerous affair fraught with risk and complication. Their overwhelming passion and naive belief that their love will persevere eventually ends in tragedy.