In this line, Juliet laments Romeo's last name as a result of the feud between their families. She asserts that his name is not as vital as a body part (hand, foot, arm, face). This line is closely akin to "A rose by any other name would smell as sweet," as both relate to the idea that one's name holds little importance in the affairs of affection.
What's more, Juliet wishes Romeo's last name to be different in the last line: "O, be some other name belonging to a man." The "O" here shows her dire discontent with the situation, and her request plays directly into the idea of her previous statement about the unimportance of title. Despite her argument that names don't matter, she still desires Romeo's name to be something other than what it is.
Juliet's musings are a reflection of her love (or lust, depending on your take) for Romeo. Her family (the Capulets) have been feuding with Romeo's family (the Montagues) for decades. She knows the two families would be enraged to know that she and Romeo were in love, and they would do anything they could to prevent them being together.
Juliet asks what it means to be a Montague--it's just a name. It's not an essential part of who Romeo is (like a hand or a face). As she states before, if was named anything else, he would still just him (and just as wonderful). She wishes for his name to be anything else, so she could be with him peacefully and not cause the trouble between their families.