Stability
A system is stable if, for all bounded inputs, the output is bounded. This means that for every $ |x(t)|\leq\epsilon $ fed into the system, $ |y(t)|\leq A $. In other words, a system is stable if all bounded input produces bounded output.
Instability
A system is unstable if there exists a bounded input that produces a non-bounded output.
Examples
Stable system
The cruise control on your car (assuming, of course, your car is equipped with cruise control) is an example of a stable system. For a given input -- a change in vehicle speed -- the system responds with a bounded, finite, and predictable output.
Unstable system
The former Tacoma Narrows Bridge is an often-used example of an unstable system. For a finite input (the wind), the bridge responded in a manner that violated its physical limits; we can assume such a response to be non-bounded or infinite, because the bridge ultimately tore itself apart.
Note
Feeding a non-bounded input into a system tells us nothing useful about the boundedness of the system, or lack thereof.
