Solubility
The solubility of a compound describes how easily it dissolves in a particular solid, liquid or gas called a "solvent." In the laboratory, the most common solvents are liquids because they are the easiest to handle.
The extent of a compound's solubility in a particular solvent depends on the interactions between the compound, called the "solute," and the solvent. If there are few interactions between the solute and the solvent, then the compound will have a low solubility in the solvent. But if there are many interactions between the solute and solvent then the solubility will be high.
Precipitation
Precipitation occurs when the products are insoluble in the solvent --- i.e. the solubility of the products is significantly lower than the solubility of the reactants. For this to occur, there must be a change in the interactions between the solute and solvent when going from the reactants to the products. Instead of relying on a change in the solute-solvent interactions caused by the reaction, you can induce precipitation by adding a different solvent to your reaction mixture.
Effect of Temperature on Solubility
Increasing the temperature of a solvent increases the solubility of a compound in a given solvent. This means, for example, that if you had 1 litter of solvent, you would be able to dissolve more of a given compound at a higher temperature than at a lower temperature. When you can no longer dissolve any more of a compound in a solution it is said to be "saturated" and the concentration will not increase.
Effect of Temperature on Precipitation
The temperature of a solution can be used to control the precipitation of a compound from the solution because the solubility of a compound in a particular solvent, and hence the saturation concentration, is dependent on the temperature.
If you lower the temperature of an iron-solution you cause the solubility of the iron compound to decrease. If the temperature is lowered enough so the saturation concentration decreases below the concentration of iron, then the excess iron forms a solid precipitate. The lower the temperature, the more precipitation of iron.