How many types of acids are there?

Acids may either be strong or weak. Besides that, there are monoprotic acids and polyprotic acids.

A strong acid is one that completely ionizes (dissociates) in a solution. In water, one mole of a strong acid HA dissolves yielding one mole of $H^{+}$ (as hydronium ion $H_3{O}^{+}$) and one mole of the conjugate base, $A^{-}$ . Example HCl

In contrast, a weak acid only partially dissociates. Examples in water include carbonic acid $\left(H_{2}C{O}_3\right)$ and acetic acid $\left(C{H}_{3}COOH\right)$