When I read my ingredients on my nail polish remover, it has water as its first ingredient. Why does nail polish have water, yet the warnings still say its flammable?
Mixtures of water with flammable liquids can themselves be flammable, largely due to the fact that the flammable liquids generally are more volatile than water, (ie they evaporate more easily.) As a result, the air above a solution of nail polish will have a lot of molecules of the flammable liquid in it. (That's why you can smell them easily). If you have a flame near the liquid, those molecules in the air above it can combust. The heat generated will evaporate more liquid, which will fuel the fire. Some of the heat is lost to vaporizing water, which doesn't burn, but if the fraction of the total liquid that is flammable is high enough, the combustion continues.
In nail polish, the flammable liquid is usually acetone or ethyl acetate.
Flammability (technically, inflammability, but I guess people are confused by this term) is a function of some vapor. Yes, water is not flammable, but the flammable liquid, in the liquid state, also doesn't burn. It's specifically the vapor that is flammable. Usually, there is some threshold concentration, reflected in the partial pressure of the flammable gas, that is required to ignite the material. Once it ignites, the heat generated by the combustion reaction promotes combustion of the adjacent vapor and heats the liquid enough to produce more gas.
If you mix water and your flammable solvent (assuming they're miscible, which they kind of have to be), you have a solution. Raoult's law says that the partial vapor pressure of the gas above the solution is proportional to the mole faction of the liquid in the solution. In this case, it means that even if the flammable liquid is diluted with water, there's still vapor above the liquid. And though the pressure may be less, it may be high enough that it's still highly flammable.