Is there a name for this set? In my book, I came across this set:
$$\{\mathbf{x}\in\mathbb{R}^n:\|\mathbf{x}\|=1\}$$
where $\|\mathbf{x}\|$ is the Euclidean norm.
Is there a name for this kind of set?

This is called the unit sphere in $\mathbb{R}^n$. It utilizes the $\ell_2$ norm, which is the standard Euclidean norm. The unit sphere for other norms will look different, for example using the $\ell_1$ norm in $\mathbb{R}^2$ gives you a diamond shape, and the $\ell_\infty$ norm gives a square.
The (closed) unit ball is a similar set, $\{ x \in \mathbb{R}^n : \|x\| \leq 1 \}$, which is a unit sphere plus all the points contained within it.
