Temperature
Basically, solubility increases with temperature. It is the case for most of the solvents. The situation is though different for gases. With increase of the temperature they became less soluble in each other and in water, but more soluble in organic solvents.
Polarity
In most cases solutes dissolve in solvents that have a similar polarity. Chemists use a popular aphorism to describe this feature of solutes and solvents: "Like dissolves like". Non-polar solutes do not dissolve in polar solvents and the other way round.
Pressure
Solid and liquid solutes
For majority of solid and liquid solutes, pressure does not affect solubility.
Gas solutes
As for gasses the Henry's law states that solubility of gas is directly proportional to the pressure of this gas. This is mathematically presented as: p = kc, where k is a temperature dependent constant for a gas. A good proof of Henry's law can be observed when opening a bottle of carbonated drink. When we decrease the pressure in a bottle, the gas that was dissolved in the drink bubbles out of it.
Molecular size
The larger the molecules of the solute are, the larger is their molecular weight and their size. It is more difficult it is for solvent molecules to surround bigger molecules. If all of the above mentioned factors ale excluded, a general rule can be found that larger particles are generally less soluble. If the pressure, and temperature are the same than out of two solutes of the same polarity, the one with smaller particles is usually more soluble.
Stirring increases the speed of dissolving
Stirring does not have an affect on solubility of a substance, but everyone knows that if he puts sugar in his tea and does not stir, it will not dissolve. Actually, if we left the tea to stand for a long enough time, the sugar would dissolve. Stirring only increases the speed of the process - it increases move of the solvent what exposes solute to fresh portions of it, thus enabling solubility. As molecules in liquid substances are in constant move, the process would take place anyway, but it would take more time.