An ionizable hydrogen is also called an acidic hydrogen. It occurs in acids, which all dissociate to some degree in water to ions like this:
HA (aq) + H2O (l) --> A- (aq) + H3O+ (aq)
The H on HA is an ionizable hydrogen. Examples include HCl (hydrochloric acid), HNO3 (nitric acid), and H2SO4 (sulfuric acid, 2 ionizable hydrogens).