While reading through some engineering related journals, I've come across the following notation:
$$\|p − Xw\|^2_2$$
$p$ and $w$ are vectors, while $X$ is a matrix. I understand that $\|p − Xw\|$ is finding the norm of the vector, i.e. $\left\|{\boldsymbol {x}}\right\|:={\sqrt {x_1^2+\cdots +x_n^2}}$. But what does two 2s mean at the subscript and superscript? I've seen them in multiple places, so I assume that it is a common notation, yet I could not find the exact definition online (it's hard to search when I don't know the terminology nor know how to type such mathematical function in google).
A norm is actually something much more general than simply the expression you gave. It is simply a function that satisfies certain properties. However, it turns out that the $2$-norm is exactly the norm you are used to.
In general, the $p$-norm of a vector is given by $$\|\boldsymbol {x}\|_p = \big(|x_1|^p + |x_2|^p + ... + |x_n|^p\big)^{1/p}.$$
You will see that by plugging in $p = 2$ you get the norm you are used to.
The superscript simply refers to ordinary squaring, hence $$\|\boldsymbol{x}\|_2^2 = |x_1|^2 + |x_2|^2 + ... + |x_n|^2.$$