Amplitude
The amplitude of a sound is physically defined as the amount of displacement produced by sound waves onto air particles. From the listener’s point of view, this variable represents the “volume” of the sound.
$$S(t) = A\sin(\omega t + \phi)$$
Given a sinusoidal function, we define $A$ to be our peak amplitude, or the highest volume that our sound reaches. That is, $\forall t \in \Reals, \space x(t) \in [-A, A] $, (for all inputs $t$, representing time in our sinusoidal function, $x(t)$ will always be between and include $-A$ and $A$).
We can define the amplitude value $A$ in one of two ways:
as a constant $a \in \Reals$, where the volume remains the same for all $t$.
as a function $A(t)$, where $t\in\Reals$, and the volume can change over time.
Using amplitude as a function of time, we can freely define how sound is outputted over time. For example, suppose we want a sample $S$ that is $n$ seconds long to fade in over the course of $m$ seconds. We can define $A(t)$ as
$$A(t) = \begin{cases} \frac{t}{m} && t \lt m \\ 1 && m \le t \le n\end{cases}$$
where $A(t)$ is a piecewise function with two parts: in the first part, $A(t)$ linearly increases from $0$ to $1$ as $t$ approaches $m$; in the second part, $A(t)$ is a constant $1$ for the remainder of the function.
Similarly, we can define a function which fades out a sound for the last $m$ seconds as
$$A(t) = \begin{cases} 1 && t \le n - m \\ \frac{n - t}{m} && n - m \lt t \le n\end{cases}$$
where $A(t)$ is constant for $t\in [0, n-m)$ and linearly decreasing for $t\in (n-m, n]$.
Lastly, we can combine both filters into a single piecewise function, giving us the effects of both:
$$A(t) = \begin{cases} \frac{t}{m} && t \lt m \\ 1 && m \le t \le n - m \\ \frac{n - t}{m} && n - m \lt t \le n\end{cases}$$
Units
In practical application, it is better to define amplitude in some physical unit of loudness rather than just an arbitrary unit. Amplitude is commonly define in terms of Decibels (dB), and the unit conversion is
$$\text{dB}(t) = \log10(\frac{A(t)}{|A_{\text{ref}}|})$$
where $|A_{\text{ref}}|$ is the greatest amplitude encountered in the entire sound. This function attempts to scale all amplitude values to the acceptable hearing range of $(-\infty, 0] \text{ dB}$, though that doesn’t mean that values above $0 \text{ dB}$ aren’t possible. Any values above $0 \text{ dB}$ represent undesirably loud values (which should be mitigated at all times, in order to protect your hearing and your audio equipment.)
For a practical example, suppose a sound $S$, normalized to amplitude values in the range $A(t)\in[-1.0, 1.0]$, contains some values that are too loud for normal listening. If we want a cutoff of $c = 0.8$ applied to the entire sound, we can substitute $A$ for
$$A_\text{cutoff}(t) = \begin{cases} c && A(t) \ge c \\ A(t) && \text{otherwise}\end{cases}$$
which will “plateau” any values above $c$, essentially rescaling amplitude while maintaining the loudness of existing values in our sound.