Signals and systems : why do we study causal signals?
An LTI system (or even a system that isn't L or TI) that is "causal" has a prayer of being realized in real time whereas an "acausal" system cannot ever be realized in real time because an acausal system responds to input from the future.  A causal system responds only to input of the present and/or input from the past.

For LTI systems, it can be shown that the output $y(t)$ depends strictly on the input $x(t)$ and the system impulse response $h(t)$.  This dependence is expressed as the operation we call "convolution":

$$y(t) = h(t) \circledast x(t) \triangleq \int\limits_{-\infty}^{+\infty} h(\tau) \, x(t-\tau) \, d\tau $$

or

$$y(t) = \int\limits_{-\infty}^{+\infty} x(\tau) \, h(t-\tau) \, d\tau $$

If and only if $h(t) = 0$ for all $t&lt;0$, then the output $y(t)$ has no dependence on future values of the input, $x(t)$.

$$y(t) = \int\limits_{0}^{+\infty} h(\tau) \, x(t-\tau) \, d\tau $$

or

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

That is the salient difference between causal and acausal systems.

Now since the impulse response $h(t)$ is called "causal" if it is zero for all time before $t=0$ (meaning the impulse response does not respond before the impulse hits the system input), that definition was extended to all signals, even signals that are not an impulse response.

The reason that it is significant is that it turns out that the real and imaginary part of the frequency response of a causal system must be Hilbert transform pairs; they must be related to each other as Hilbert transforms of each other.