A common distillation with an azeotrope is the distillation of ethanol and water. Using normal distillation techniques, ethanol can only be purified to approximately 96% (hence the 96% (192 proof) strength of some commercially available grain alcohols).
Once at a 96.4% ethanol/water concentration, the vapor from the boiling mixture is also 96.4%, therefore further distillation is ineffective. Some uses require a higher percentage of alcohol, for example when used as a gasoline additive. The 96.4% azeotrope needs to be "broken" in order to refine further.