The impulse response of a discrete-time system is the output produced by the unit impulse:

$$h[n]=T{\delta[n]}.$$

For an LTI system, this one sequence determines the response to every input.

A general input can be decomposed as

$$x[n]=\sum_k x[k]\delta[n-k].$$

Time invariance means the response to $\delta[n-k]$ is $h[n-k]$. Linearity means the response to the weighted impulse $x[k]\delta[n-k]$ is $x[k]h[n-k]$, and the responses to all terms can be added.

The impulse response is therefore a complete time-domain description of a discrete-time LTI system's zero-state input-output behavior. Convolution is the explicit sum obtained from this superposition.