Solubility
Solubility is the maximum amount of a solute that can dissolve in a given amount of solvent at a specific temperature and pressure.
Definition
Solubility is how much of a substance can dissolve in a liquid: some things dissolve a lot, some only a little, and some not at all. It is defined as the maximum mass of solute that will dissolve in a fixed amount of solvent (often $100\text{ g}$ of water) at a given temperature, and a solubility curve plots this limit against temperature; the solubility of most solids increases with temperature, while the solubility of gases decreases with temperature and increases with pressure. At equilibrium, solubility is the concentration of a saturated solution, governed by the free energy change of dissolution, $\Delta G = \Delta H - T\Delta S$. For sparingly soluble ionic salts it is expressed through the solubility product constant $K_{sp}$, for gases Henry's law states that solubility is proportional to the partial pressure above the liquid, and temperature dependence follows the sign of the enthalpy of solution, so endothermic dissolution becomes more favorable as temperature rises.
Example
You can stir a lot of sugar into water before it stops dissolving, but only a little salt will dissolve in the same water, and sand does not dissolve at all. At $20^\circ\text{C}$ about $36\text{ g}$ of sodium chloride dissolves in $100\text{ g}$ of water and about $204\text{ g}$ of sugar dissolves in the same amount, while potassium nitrate jumps from about $32\text{ g}$ at $20^\circ\text{C}$ to about $246\text{ g}$ at $100^\circ\text{C}$. Silver chloride has $K_{sp} = 1.8 \times 10^{-10}$ at $25^\circ\text{C}$, and since $K_{sp} = [Ag^+][Cl^-] = s^2$, the molar solubility is $s = \sqrt{1.8 \times 10^{-10}} \approx 1.3 \times 10^{-5}\text{ M}$, or about $1.9\text{ mg}$ per liter.
Key Insight
Warm water usually dissolves more solid than cold water, which is why sugar dissolves faster and more completely in hot tea than in iced tea. Gases behave the opposite way: a warm soda goes flat faster because carbon dioxide is less soluble in warm water, and a lake that warms in summer holds less dissolved oxygen for fish. The common-ion effect lowers solubility, so adding $NaCl$ to a saturated $AgCl$ solution forces more silver chloride to precipitate, and kidney stones, water softening, and scale in boilers are all solubility equilibria in action.