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

Good for you, Maryam Diaab; for being extra safe.
You will want a nice smooth surface and the barres mounted into studs with long bolts. Michael Siler’s idea of contacting a contractor is a good idea; but, I would still go with more than the minimal requirements.
Then of course, just like Michael Saiz says, different situations come into play, also.
Doing things the right way the first time saves more than a stitch in time.
Take care.