Mineral spirits are organic solvents derived from the petroleum distillation process. They are typically long chain hydrocarbons. They are excellent for thinning enamels, cleaning airbrushes after using enamels, and enamel or even oil washes.
Lacquer thinners are also organic solvents but are more severely refined products; they may contain xylene, toluene, and other similar solvents. These materials are typically ring-based (ie, benzene) organic structures. They are "hotter" than mineral spirits. Use this thinner for lacquers, but also for cleaning out your airbrush.
Acrylic thinners also use organic solvents, but the key is that they are water soluble. Materials include isopropyl alcohol, and (I suspect) some glycol ethers. Use these thinners for spraying, washing, and cleaning your airbrush after using acrylics. You may also use rubbing alcohol (70% isopropyl), windshield washer fluid (methanol), and even tap water with a drop of dishwashing liquid as acrylic thinners. My advice would be to use the recommended manufacturers' acrylic thinners first (if you can get it), and then experiment with the other stuff to see what works best for you.
Hope this helps a little
Gip Winecoff