Stability is a measure of the tendency, or lack thereof of a the floor to expand and contract across its width with changes in relative humidity. All wood is hygroscopic, meaning it reacts to the presence of moisture in the air around it. Stable woods do not change as much in width as a less stable wood.
Quarter sawn lumber does not react across its width due to the direction of the growth rings and so expansion and contraction issues (gapping) can be all but eliminated using quarter sawn flooring products.
The use of a whole house humidifier in the heating months, and air conditioning during the summer months will also limit or eliminate expansion and contraction issues. Maintaining a consistent and proper humidity level is also better for you and your family, your fine furniture, and houseplants.
When using wider planks in less stable species without humidity controls there are a number of techniques a qualified installer can employ to limit the aesthetic and structural significance of the eventual movement.
The chart below compares the propensity of a number of common species, a lower number indicates a more stable floor.