Stability refers to the amount of arch support that a shoe offers.
One common method of assessing the height of your arches is to look at a wet footprint, but of course this is only one thing to consider.
People with lower arches would tend to see more roll in the foot, which is called pronation. This is a perfectly normal action of the foot, but most runners who overpronate will benefit from a shoe with added stability on the inside of the arch.
The amount of stability you will benefit from will also depend on the distance you run, and your training history.