The impulse response of a continuous-time system is the output produced by a Dirac impulse:

$$h(t)=T{\delta(t)}.$$

For an LTI system, this one function determines the zero-state response to every input.

A continuous signal can be represented formally as

$$x(t)=\int_{-\infty}^{\infty}x(\tau)\delta(t-\tau),d\tau.$$

Time invariance makes the response to $\delta(t-\tau)$ equal to $h(t-\tau)$. Linearity lets the weighted responses be superposed, giving

$$y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau),d\tau.$$

The impulse response is therefore a complete time-domain description of continuous-time LTI zero-state input-output behavior. The resulting integral is continuous-time convolution.