What are the backboards that some ballet barres mount to? Are they regular planks of wood or something special?

Hello Maryam,
That would depend on the type of barre wall bracket hardware being used:
Here are some helpful links:

http://www.alvas.com/barre_adj_brackets_page.html

http://www.alvas.com/wall_mount_adjustable_ballet_barre_installation_ins…

http://www.alvas.com/barre_fixed_brackets_page.html

http://www.alvas.com/wall_mount_fixed_ballet_barre_installation_instruct…

http://www.matsmatsmats.com/ballet-dance/ballet-bar-wall-mounted.html

I Hope This Helps!
Michael