The word most frequently used by far in American English by children to refer to other children is kid itself.
I met this kid at the playground today.
I'm sitting next to a new kid in class this week.
After that, gender-specific terms to use by a boy would be girl (to refer to a girl) or boy by a girl to refer to a boy. This would be to distinguish the gender because any anonymous child referenced would be assumed to be the same sex.
There are no built-in pejoratives. You'd have to add an adjective or use another noun: "That stupid kid" or "that dirty boy" or "idiot" (depending of course on what traits the speaker wanted to single out).
The one word practically never used by a child to refer to another child would be child. Same with children.
There's a bunch of new kids in my class. [not children]
Addendum
Having had this question in the back of my mind for the better part of a day, I feel obliged to observe that "kid" is probably a bit less likely to be used when referring to a girl by another girl. Not that it couldn't be, but in American English, as in many languages, generic terms seem to have a default "maleness" about them. I don't want to get into the politics of that here, and do recognize that I say this only as an observer, not a proponent. But there it is. In the comments to this question, Brian observed a correspondence between "kid" and "guy" and I do endorse that similarity. I would extend it further to cover this kind of vague male undertone to the word, as I have done in the past. I think the polarity is possibly less with "kid" than with "guy," but I could be wrong.
Best Answer
This would be called poster board. It may also be called a tri-fold display board.