Acoustic phonons are basically the phonons you can in a lattice where there is only 1 ion. Say maybe a pure (ie. contains only) aluminium lattice.

 

Whereas in a lattice where you get 2 ions, say sodium chloride (aka salt) which contains both sodium and chloride. As there are two different ions there are two different modes of vibration... one of these is the "acoustic" vibrations (quantised with acoustic phonons) and the other is "optical" vibrations (quantised with optical phonons).

 

http://www.chembio.uoguelph.ca/educmat/chm729/Phonons/optical.htm

The allowed frequencies of propagation wave are split into an upper branch known as the optical branch, and a lower branch called the acoustical branch. There is a band of frequencies between the two branches that cannot propagate. The width of this forbidden band depends on the difference of the masses. If the two masses are equal, the two branches join (become degenerate) at . Note that the first Brillouin zone goes from to just as in the monatomic case if we use the lattice constant instead of the interatomic spacing . The acoustical branch is qualitatively similar to the dispersion relation for a monatomic lattice, but the optical branch represents a completely different form of wave motion.

How excellent, thank you. Somewhere (35 years ago) I heard "Brillouin" zone but I never got grounding in solids.