Acid-base reaction: Al(OH)₃ + NaOH
All reaction sources: Chemiday

As I said in the comment section that aluminium hydroxide is amphoteric i.e. It dissolves in acid, forming $\ce{[Al(H2O)6]^3+}$ (hexaaquaaluminium) or its hydrolysis products and dissolves in strong alkaline solutions, forming $\ce{[Al(OH)4]−}$ (tetrahydroxidoaluminate). So, when it is dissolved in base, it forms a sodium aluminohydroxo complex salt. Why? Because as @Pritt mentioned, aluminium has vacant orbital for which free electron can reside there forming a complex salt.

$$\ce{Al(OH)3 + NaOH(conc.) → Na[Al(OH)4]}$$

$$\ce{Al(OH)3 + 3NaOH → Na3[Al(OH)6]}$$

No worries. There is also an acid-base reaction where the two combines to form a salt and water but this occurs at very high temperature and you won't find this type of reaction occurring in laboratory. This reaction can only be find in industries.

$$\ce{Al(OH)3 + NaOH -&gt;[1000 C] NaAlO2 + 2H2O}$$

If only aluminium was used, there would be evolution of hydrogen gas from a very similar exothermic reaction. Note for simplicity, the species is written  $\ce{NaAlO2}$ but it actually contains $\ce{[Al(OH)4]−}$ ion or perhaps the $\ce{[Al(H2O)2(OH)4]−}$ all due to presence of vacant orbital(see sodium aluminate).