The solubility of the compound X is the thermodynamic term. The solubility of X is the concentration of X in saturated solution regardless a phase of X (solid, liquid, or gas). A supersaturated solution is thermodynamically unstable.
If a compound dissolves in a solvent it is soluble. Solubility is a function of temperature; some substances experience a higher solubility as temperature increases, others, like sodium chloride in water, have a small increase in solubility as temperature increases. Finally some substances can present a decrease of solubility in a certain range of temperature, like calcium carbonate above 100 degrees.
the relation you asked is: solubility is the maximum concentration of a substance that dissolves at a certain temperature. The variation of concentration with the temperature is the solubility curve. If the concentration is lower than this value, the solution is undersaturated. If the solution is at a maximum concentration, the solution is satured. The concept of supersaturation is due an histeresis that the solubility suffers: when you cool a undersatured solution, at a certain temperature the solution is saturated. Cooling more the same solution bellow the temperature of saturation it continues in just one phase, metastable and supersatureted, due the energy necessary to form a surface and volume of solids. After a certain time the solid phase is formed and the solution tend to grow the solid phase until a new point in the equilibrium curve is reached and crystallization finishes.
Solubility of a substance is calculated usually at 20 degree, and depends of temperature and presure, if you want to get a saturated or suprasaturated solution you can heated the solution and then let to could at 20 degree, will see that the excess of substante will crystalizes. If is not soluble at 20 degree it is must stated that.