A set of groups is stable if the difference between Black and White making the next move near to it has a small value.
The book Joseki Volume 2 Strategy by Robert Jasiek defines on p. 94 carefully in detail:
"A set of nearby groups is stable if all these properties apply:
- Each group is alive unless it is a proto-group or a sacrifice.
- At the boundaries of the live or unsettled black and white stones, neither player has any local play that is currently sente and does not just waste aji or that would make a huge difference compared to the opponent playing there first. Similarly putting helping stones in front of a wall in sente is impossible.
- The stones of each live group are connected.
- None of the live groups can take another major development direction.
- Both players have either no or about equal options."
Robert Jasiek explains on p. 88 of Fighting Fundamentals that many kyu players are not well aware of the concept of stability and states this principle:
Important groups should maintain stability.
Since important groups also need life, one can even say that they should maintain both life and stability. Strong dan players apply this principle consistently, unless they want to sacrifice life or stability of a group for advantages elsewhere.
"Stable eye" is a different term with a similar name (Definition from Capturing Races 1, p. 31 by Robert Jasiek):
"An eye is stable if, supposing the player surrounding the eye and his opponent play only within the eye's region, the player moving first cannot necessarily permanently partition it into at least two regions."