Stability is a characteristic that we desire in all technology. Without stability, things get precarious; we become unsure of whether things will break at any moment. Stability is best exemplified by a common piece of real-world technology. Compare these two bridges:

They both technically operate correctly as bridges. However, one has previously suffered damage and has been fixed with a simple plank of wood. Which bridge would you trust to safely convey a hundred people across? Probably the one on the right. It is firmly in place with guard rails, and, crucially, no gaps through which you can fall. 

In code, we can say that stability is about the continued correct behavior given different valid inputs and situations. JavaScript in the browser is especially liable to failures in this way. It has to run in a multitude of conditions, on different hardware, operating systems, screen sizes, and often in browsers that have varying capabilities.

If we write code that is strongly dependent on certain conditions, then it may be unwieldy and undependable when those conditions do not exist. If, for example, we were designing and implementing a layout for a web application, we may forget to consider and cater for screen sizes less than 1,024 pixels wide, resulting in the following mess:

This is an example of instability. The web application cannot be depended upon to deliver its correct behavior when a certain environmental factor is different. A situation in which screen size is less than 1,024 pixels is entirely possible and reasonable in a world of increasing mobile device usage; it's an absolutely valid use case of our web application and failure to accommodate it has a negative effect on our user's ability to rely on it.

Stability is gained through having a full understanding of all the different valid inputs and situations that your code may be exposed to. Similar to correctness, stability is best confirmed with a set of tests that expose your code to the full gamut of inputs and situations.