Stability dictates the Stamina that is drained whenever one is hit whilst blocking and thus determines how long one can continually block while receiving blows.
Certain classes of shields and weapons have higher stability than others. Greatshields are known for having the highest stability out of the shields, with the highest being a Crystal Tower Shield, while Ultra Greatswords have the highest out of the weapons.
While reinforcing a shield usually increases its stability, some shields only have their damage and elemental resistance increased. Elemental resistance can only be upgraded by ascending a shield down the matching elemental path. An increase in damage mostly occurs on Unique Shields.
Apart from reinforcing, there is another way (although temporary) with the Magic Shield & Strong Magic Shield sorceries, both last 15 seconds (~22 with the Lingering Dragoncrest Ring) and grant an additional stability of 18 and 34 respectively. Since this stat acts as a percentage of the stamina you will not lose, any shield with 82 stability will block all attacks without stamina loss, and the same for shields with 66 only using the strong version.