Julian Carlson is Rachel's son, and Juliet's nephew, who was born while Juliet was on the Island. Juliet sees him for the first time in a video filmed by Richard shown to her by Ben and Mikhail in the Flame, to prove to Juliet that Rachel was still alive and had successfully given birth to a boy. It is probable that he is named after Juliet.

Unanswered Questions
Who is his father?
Is there a father, or is his birth the product of the treatment administered to Rachel by Juliet?

I wonder if we will ever find out more about how he was conceived.

For all we know - he is not even her child. that whole scene that Richard watched could have been staged....

That is true. It could be just a random kid as well.

So much of the Others world is smoke and mirrors. Reflecting back on the "ease" of showing this to Juliet, I suspect we will learn that this happy showing of mother and son togetherness was more than it appears.

I agree, I wouldn`t be surprised either if it turns out there is something more to this story.