It is a graph of how much of a substance (the solute) can be dissolved in a solvent at various temperatures.
The x-axis will have Temperature values and the y-axis will have the amount of solute dissolved, usually in units of grams per 100 grams of the solvent. Most often you will find graphs with water as the solvent. The purpose of the graph is to show how much of a solute can be dissolved to make a saturated solutions.
If we think of this in another way, a solubility curve shows the concentration of a solute that will result in precipitation of the solute. This happens when there is more solute present than what the solvent can hold in solution. (An exception is when a supersaturated solution is made, but this situation is very unstable.) For example, if a saturated solution cools or the solvent evaporates, there is not enough solvent to keep all the solute dissolved.
Here is a link with some further explanation and an example of such a graph:
A search of the Internet will find lots of other examples.
As you examine various examples you will probably notice that as temperature increases, the solubility of many solid substances increase while gases become less soluble.