Transient response can be directly related to Natural Response & steady state response can be directly related to Forced response. Initially the response will be mixed one & after some time natural response will die out & only forced response of system will be there which will be dependent upon input.
Consider a standard spring-mass system & suppose there is no damper in the system & system is disturbed from mean position & released, system will never return to its mean position. It will perform vibrations about mean position. System is said to be unstable in this case. Now suppose there is damper in the system, the system will return to mean position after some time which will be dependent on amount of damping.
So for system to be stable, transient response should die out after some time.Once the transient response dies out, the systems response will depend on input only.
If the input is bounded one then output will be bounded one.This leads to another definition of stability. System is said to be stable if bounded input to system gives bounded output(BIBO).
Non-linear systems have Asymptotic stability means their output reaches to equilibrium as 
Linear systems are only exponentially stable means their output reaches to equilibrium in exponential fashion.
