Molarity and molality are both measures of concentration of solutions. One is the ratio of moles to volume of the solution and the other is the ratio of moles to the mass of the solution.
Molarity, also known as molar concentration, is the number of moles of a substance per liter of solution. Solutions labeled with the molar concentration are denoted with a capital M. A 1.0 M solution contains 1 mole of solute per liter of solution.
Molality is the number of moles of solute per kilogram of solvent. It is important the mass of solvent is used and not the mass of the solution. Solutions labeled with molal concentration are denoted with a lower case m. A 1.0 m solution contains 1 mole of solute per kilogram of solvent.
For aqueous solutions (solutions where water is the solvent) near room temperature, the difference between molar and molal solutions is negligible. This is because around room temperature, water has a density of 1 kg/L. This means the "per L" of molarity is equal to the "per kg" of molality.
For a solvent like ethanol where the density is 0.789 kg/L, a 1 M solution would be 0.789 m.
The important part of remembering the difference is
molarity - M → moles per liter solution
molality - m → moles per kilogram solvent
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