A group is stable when its life and death status is very unlikely to change. A position is stable if the addition of a stone doesn't significantly alter the position.
Stable could be a useful synonym for thick. In fact, I (Dieter) have heard it used consistently by a Japanese ex-insei who very probably had never read English Go literature. He thus translated atsumi as stability.
To interpret the above: adding a stone to a dead group is a waste. So is adding a stone to a living group. Adding a stone where the opponent is strong is not useful. So is adding a stone to a position where you are strong. Play away from thickness or as the insei said: don't play close to stable positions.
exswoo Would I be correct in assuming that thickness would actually be a particular case of a group of stones being stable? For example, I don't think a stable black group that is surrounded by white stones would be considered thick...
...or do you want to reword the definition of stable slightly so that it encompasses this? ;)