We don't know much about the narrator's baby, but it plays a very important symbolic role in the story, nonetheless. It's instructive in this regard that the unnamed woman has been forced to convalesce in a room that had originally been set aside as a nursery. This drives home the point that the woman is being infantilized by her husband, treated like a small child rather than the grown woman she is. In her present condition, the narrator is little more than a helpless baby, utterly dependent on her physician husband for emotional support. However, such support is not forthcoming, and so the narrator is as vulnerable as a baby who's been left to fend for itself.
The baby also serves to heighten the narrator's separation from the world around her, making her more prone to descending into a world of pure fantasy. Physically separated from her newborn child, who's...
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being taken care of by a nanny, the narrator has been deprived of any connection to reality. The birth of a child, which should be a happy, joyous occasion, has instead precipitated a tragic descent into mental illness. In that sense, the baby, though kept out of sight, is the unwitting catalyst for everything that subsequently happens in the story.
As the other answer indicated, we know that the narrator is separated from her baby, who is being taken care of by another woman, possibly a nurse:
It is fortunate Mary is so good with the baby. Such a dear baby!
However, it sounds as if she might like to see him, but is repeating what she has been told:
And yet I cannot be with him, it makes me so nervous.
We also learn that the baby would have been put in the nursery with the yellow wallpaper if the narrator had not been placed there instead:
If we had not used it, that blessed child would have! What a fortunate escape! Why, I wouldn't have a child of mine, an impressionable little thing, live in such a room for worlds. I never thought of it before, but it is lucky that John kept me here after all, I can stand it so much easier than a baby, you see.
We see in the passage above that the narrator wants to keep the baby safe. We don't learn much about the baby himself, but we do know he is separated from his mother, that he is being cared for, and that the narrator has protective instincts towards him.
Beyond that, the baby doesn't seem to have much reality to the mother. She doesn't describe him in terms that would differentiate him from any other baby.
Readers find out relatively little about the baby in "The Yellow Wallpaper". As we're dealing with an unreliable narrator—someone we can't fully trust due to them being misguided, confused, or straight-out lying—one could argue we really don't know anything about the baby at all.
If we do go by the words of the narrator, most of what we can tell about the child is how he makes her feel. That, in turn, actually says a bit more about the mother. The passages that refer to the baby hint that unlike his mother, the boy is happy and healthy:
There's one comfort, the baby is well and happy and does not have to occupy this nursery with the horrid wallpaper.
This brings out something that the short story refers to often at the beginning of the tale: the narrator, while she regrets being weak and ill, is glad the child is not with her. Her feelings are conflicting. On the one hand, she can't bear to see him or be with him much, yet she also misses him and wants to be better. But she definitely makes no attempt to have the boy brought to her, into the terrible room. So in a way, she shows protectiveness over her son. The narrator continues, explaining how she can bear the mood of the room so much better than her infant child would be able to.
In bits and pieces, everything we find out about the baby is therefore pretty expected. Both his parents are trying to do what's best for him, however misguided these actions may be (in the case of his father). He is well taken care of, loved by his parents and in good health—other than that, he is simply a newborn boy.
The narrator of the story says relatively little about her baby. She does, at one point, say,
It is fortunate that Mary is so good with the baby. Such a dear baby! And yet I cannot be with him, it makes me so nervous.
From this reference, we can glean a few pieces of information. First, the baby is a boy. Second, the narrator does not spend much, if any, time with the baby because being around him makes her too nervous. Third, the baby has a governess or nanny named Mary, someone who takes care of him in his mother's absence. Fourth, this description of how the baby makes her feel, in addition to a few other descriptions of her "nervous condition," helps us to understand that she is likely suffering from what we'd now call postpartum depression, for which there was no name then. What we know now as postpartum depression was simply lumped under the catchall diagnosis of hysteria, which is one of the words the narrator uses when she describes her physician husband's opinion of her illness.