What does this notation $g(x) = S\{\delta(t) \}(x)$ mean? What does this notation mean $g(x) = S\{\delta(t) \}(x)$ mean, where $S$ is a "system", $\delta(t)$ is the impulse function and $g(x)$ the output function of the system? I am really not familiar with signals and signal processing. I know the definition of the impulse function 

$$
\delta(t) = 
\begin{cases}
1, \; t = 0,\\
0, \; t \neq 0
\end{cases}
$$

I understood that a system is a function that gets as input a function and returns another function (so it should be a functional). However, I do not get this notation $g(x) = S\{\delta(t) \}(x)$? For example, why the parentheses $\{ \}$? Why do we apply the $\delta(t)$ inside $S\{ \cdot \}$? What is the precedence of those function applications? How should I read that formula? Why do we even use an impulse here, what is the intuition behind this?

In my case, also have that 

$$g(x) = S\{\delta(t) \}(x) = (A\delta(t))(x) = A_{x, 0}$$

Why are the notations $S\{\delta(t) \}(x)$ and $(A\delta(t))(x)$ equivalent? What is this $A_{x, 0}$? $A$ should be a Toeplitz matrix, but I don't understand why. How should this matrix $A$ look like?

$A_{x, 0}$ should represent coefficient associated to instant $0$ for computing the output at instant $x$. I also don't get this, even though I think we are trying to convert a continuous signal to a discrete one. 

You can assume that $S$ is a linear shift-invariant system.
My understanding is that a system $S$ is a notation for   a generic object, which takes instances depending on its arguments, given between $\{\cdot\}$. Here, $S\{\delta(t)\}$ instantiates it as its effect on the zero-centered impulse response, with free variable $t$, taken on variable $x$. 

Now, if this system $S$ is linear and shift-invariant, the operator given by $S\{\delta(t)\}$ defines $S$ it completely. 

If the system applies on discretized inputs (that we can call $x$), it can be rewritten as a linear operator, represented by an infinite Toeplitz matrix. Each row implements the (linear) convolution, and the Toeplitz structure emulates the shift-invariance. I suggest to read Toeplitz and circulant matrices by R. Gray. The $0$ in $A_{x,0}$ may refer to the way an infinite matrix is matched with a signal centered at $0$, but I still wonder if this notation is clear.