Temperament: The manner of thinking, behaving, or reacting characteristic of a specific person *or in this case, Animal*
Dogs, just like people, have unique personalities that include their behavior, natural and learned traits and how they react to situations, stimuli, people and other animals.
Each breed of dog has a general guideline as to the traits that a stable dog of that breed should exhibit. Temperament is more then just traits, it is the accumulation of training, socialization, personality, history, genetics, traits and also current situations. Every dog is different, and while it is acceptable to look for these breed specific traits, one needs to consider all of these things as a whole when understanding their dog's temperament.
When you understand your dog's personality and how they react to things, training, socialization and introductions can be easier and you can get a clearer grasp on what your dog needs and how to approach meeting those needs.
Temperament tests are often used by rescues and shelters to judge the adoptability as well as the individual personality of new dogs. These evaluations are used as a tool in understanding a new dog and to identify what needs the dog has.
What do they mean by a "Stable" dog?
A stable dog is one who exhibits adaptability and tolerance to new situations. One that has breed specific traits and reacts favorably under stress. One who is friendly, outgoing and accepting.
You can use Temperament to determine what dog would be the best fit for your family. If you have children, you might consider a dog who is tolerant, calm and accepting of loud noises, lots of chaos and being touched hugged or pestered. If you have an active lifestyle you might consider a high energy dog who reacts favorably to new situations, new people and is not bothered by change.
Keep in mind that the dog's age does play an important role. Many stable puppies can still grow up having unstable temperaments if not given proper socialization, training and boundaries. Puppies Temperament changes as they enter into adulthood, sexual maturity and as they mature in general so by the time they are a mature adult, their temperament is pretty consistent. *This is why i recommend adopting adult dogs, their personality will be generally consistent unless they are placed in extremely distressing situations*