OK let's see if this helps you. Suppose you have two functions $f,g:[a,b]\to \mathbb{R}$. If someone asks you what is distance between $f(x)$ and $g(x)$ it is easy you would say $|f(x)-g(x)|$. But if I ask what is the distance between $f$ and $g$, this question is kind of absurd. But I can ask what is the distance between $f$ and $g$ on average? Then it is $$ \dfrac{1}{b-a}\int_a^b |f(x)-g(x)|dx=\dfrac{||f-g||_1}{b-a} $$ which gives the $L^1$-norm. But this is just one of the many different ways you can do the averaging: Another way would be related to the integral $$ \left[\int_a^b|f(x)-g(x)|^p dx\right]^{1/p}:=||f-g||_{p} $$ which is the $L^p$-norm in general.

Let us investigate the norm of $f(x)=x^n$ in $[0,1]$ for different $L_p$ norms. I suggest you draw the graphs of $x^{p}$ for a few $p$ to see how higher $p$ makes $x^{p}$ flatter near the origin and how the integral therefore favors the vicinity of $x=1$ more and more as $p$ becomes bigger. $$ ||x||_p=\left[\int_0^1 x^{p}dx\right]^{1/p}=\frac{1}{(p+1)^{1/p}} $$ The $L^p$ norm is smaller than $L^m$ norm if $m>p$ because the behavior near more points is downplayed in $m$ in comparison to $p$. So depending on what you want to capture in your averaging and how you want to define `the distance' between functions, you utilize different $L^p$ norms.

This also motivates why the $L^\infty$ norm is nothing but the essential supremum of $f$; i.e. you filter everything out other than the highest values of $f(x)$ as you let $p\to \infty$.

Let $f\colon A \to B$ be a function. This means $f$ takes as an argument elements of the set $A$ and returns elements of the set $B$.

For example, $g(x) = x^2$ usually is a function that takes and returns real numbers, thus $g\colon \mathbb R\to\mathbb R$.

Usually the graph of a function $f$ can only be sketched, if the sum of the dimensions of $A$ and $B$ is less or equal to 3.

In order to graph functions that map from $\mathbb R$ to $\mathbb R$ you can use e.g. desmos.

Now you were originally asking about a function $\|X\|^p$. Your notation implies that $X$ is supposed to be a vector. As I said before you can only graph a functions if the sum of the dimensions is less or equal to 3. This means you can graph $\|X\|^p$, if $X$ is two-dimensional. You can use this in order to graph it in this case. Note that it cannot handle the norm directly. The usual norms for two dimesional vectors are \begin{align*} \left\|\begin{bmatrix}x\\y\end{bmatrix}\right\|_q &= \left(|x|^q + |y|^q\right)^{\frac1q} \end{align*} for some $0<q<\infty$. You can thus graph it by typing

( (abs(x)^1.5 + abs(y)^1.5)^(1/1.5) )^5

into the graphic 3D plotter from above. It will give you a plot of $\|X\|_q^p$, where $q=1.5$ and $p=5$.