Molarity is the number of moles of solute per liter of solution, while normality is the measure of concentration qual to the gram equivalent weight per liter of solution. Normality and molarity are both measures of concentration in chemical solutions.
Molarity is the most common measurement of chemical concentration. If a solution of H2SO4 has a molarity of 1 (expressed M1), then there is one mole of H2SO4 in every liter of that solution. Normality measures the gram equivalent weight in every liter of solution. Gram equivalent weight is a measure of the reactive capacity of a molecule.