Take some sugar and drop it into coffee or tea. Stir it up and the sugar disappears. This disappearance relates to the solubility of sugar --- that is, its ability to dissolve, the speed at which it dissolves and the amount that will dissolve in a given quantity of liquid. The measure of how much sugar is in a given amount of liquid, or its concentration, is called molarity.
Solubility relates to the creation of a solution; two substances that become one. The substance that is being dissolved, normally the smaller substance, is called the solute. Sugar when placed in coffee is a solute. The large substance is the solvent, like coffee. Water is a frequent solvent. Solubility relates to the relative strengths of the solute and solvent. The easier it is for the solute to break apart, the greater the solubility.
Sugar, or C12H22O11, is a solid held together by bonds between the molecules. Those bonds represent weak intermolecular forces. When the sugar as a solute mixes with the solvent water the bonds between the molecules break and the sugar dissolves. This is a result of the interaction between the molecules of the solute and solvent and the energy generated. This process continues up to the point where 1,800 grams of sugar dissolves in one liter of water. To measure the concentration use moles of solute per liter of solution; the answer is molarity.
Molarity, notated as M, or moles per liter, is measured by the number of moles of solute divided by a liter of the solution. The mass of a solute is normally expressed in grams and must be converted to moles. That requires the use of a conversion rate, or number of grams per mole, which differs for each solute. One mole equals the atomic weight of the solute. A simple example is carbon dioxide, or CO2. Add the atomic weight of carbon plus twice the atomic weight of oxygen and the total is the number of grams of carbon dioxide in one mole.
Bodner Research Web lists three solubility rules for salts which predict whether a particular salt will dissolve in water. They involve use of molarity as a measurement of concentration. Salts with a minimum concentration of 0.1 M dissolve in water at room temperature. No dissolution in water at room temperature takes place for salts at less than 0.001 M. Solutions between the two extremes show slight solubility.