What is chemical formula of Bauxite?
$\ce{Al2O3.nH2O}$ is just a general (and slightly incorrect, see below) way of writing the formulae of the aluminium oxy-hydroxides that occur in bauxites. These are solid minerals, so there is no "$\ce{Al2O3}$" molecule in them. Instead, it is an (almost) infinite array of Al and O ions in a three dimensional structure. Therefore, what it basically means is that you have three cations of aluminium for every two anions of oxygen. $\ce{Al2O3}$ means exactly the same as $\ce{AlO_{1.5}}$ or $\ce{Al4O6}$ in this context. We just use $\ce{Al2O3}$ because it's simpler: smallest numbers with no fractions.

In the case of the hydroxides, they are also crystalline minerals. You can see some nice pictures of gibbsite, böhmite and diaspore when they are well crystallised. Their formula are:


Gibbsite: $\ce{Al(OH)3}$ which is the same as $\ce{AlO_{1.5}.1.5H2O}$
or $\ce{Al2O3.3H2O}$.
Böhmite and diaspore: $\ce{AlO(OH)}$ which is the same as $\ce{AlO_{1.5}.0.5H2O}$ or $\ce{Al2O3.H2O}$


Bauxites usually contain mostly gibbsite, which is unfortunate economically because it contains the most water in it. When you dig the stuff out, you want the most aluminium per tonne of material, and dilution by water is undesirable.

It's also important to note that even though we write the formulae of the minerals as $\ce{Al2O3.nH2O}$, the water is not hosted in molecular form in the crystal. You do not have proper $\ce{H2O}$ molecules in the structure of the mineral. Instead, you have hydrogen cations attached to some of the oxygen anions. There are, however, minerals where the water is in molecular form. One example is gypsum: $\ce{CaSO4.2H2O}$.

Usually, as a rule of thumb, the way a formula is written can give you an idea on how the water is hosted. If the formula is written with $\ce{(OH)}$ in it, it's probably in the form of hydrogen bonded to oxygen. If it's $\ce{.H2O}$ then you probably have molecular water. This is also important for the physical properties of the crystals. Minerals with molecular water are usually very soft and will lose their molecular water with gentle heating (up to 150 °C or so). Minerals with hydroxide groups in them require much higher temperatures to de-volatilise, around several hundreds of °C.

So, to sum it up, writing the formula as $\ce{Al2O3.nH2O}$ gives the impression that the aluminium hydroxides have molecular water in them, while they actually do not. However, it's a convenient "catch-all" way to refer to all of them together.

Also related:
What is the meaning of the &quot;dot&quot; notation in chemical formulas?