Mixing bleach with vinegar causes a chemical reaction that releases chlorine gas, which can be very dangerous. Chlorine gas attacks the mucous membranes, causing painful burning in the eyes, throat and lungs. Inhaling chlorine gas can cause pulmonary edema, which can, in severe cases, be fatal.
Chlorine bleach contains hypochlorous acid, which is a strong oxidizer. Mixing hypocholorous acid with another acid, such as the acetic acid found in vinegar, lowers the pH of the bleach, which increases its oxidizing power. Adding vinegar to bleach does make it a more powerful cleaner, but with the side effect of releasing high levels of chlorine gas. The health risk from chlorine gas is so great that it was used as a weapon during World War I, and more recently by insurgent groups during the Iraq War.