Intuition behind impulse response terms in convolution
The coefficient $h[n]$ is the value of the system's response at time $n$ when the input signal was an impulse at $n=0$. Obviously, that's why we call $h[n]$ the impulse response. From this you can see that for a system to be causal, $h[n]$ must be zero for $n&lt;0$, otherwise the system would "know" in advance that an impulse will come at $n=0$.

Note that any discrete-time signal $x[n]$ can be written as a sum of unit impulses:

$$x[n]=\sum_{k=-\infty}^{\infty}x[k]\delta[n-k]\tag{1}$$

Since the response to a shifted impulse $\delta[n-k]$ is $h[n-k]$, and since the system is linear and time-invariant, from $(1)$ the output signal must be given by

$$y[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]\tag{2}$$

which is of course the discrete-time convolution. Consequently, positive indices of $h[n]$ correspond to the memory of the system, that's why $h[1]$ is multiplied with the past input sample $x[n-1]$, etc.