Although there is a range of usages and meanings, these are the most common: "Normal" is usually an adjective. "The normal temperature in summer here is 30°C." "Norm" is the usual noun. "Sometimes it gets hotter, but 30°C is the norm." "Normality" is the state or condition of being normal. The "-ity" suffix means the same thing as "-ness." You just have to check a dictionary to see which form is used. "Normality" means "normalness" (but "normalness" isn't a word). "I'm tired of unusual weather! I want normality!" "Normal" is a very common word. "Norm" and "normality" are rare. "Norm" often refers to social customs. "We have laws, but we also have norms. People are expected to follow the norms even though no law says that they have do."