In $ \S $1 of Jean-Pierre Serre's article Géométrie algébrique et géométrie analytique, he uses notation that I have not seen in Hartshorne or Vakil or anywhere else: $\mathscr H_{x, U}$, where $\mathscr H$ is the sheaf of germs of holomorphic functions on $\mathbb C^n$. In my mind, $\mathscr H_{x, U} = \mathscr H_x$, and yet (as seen in the excerpt below) Serre talks about the kernel of $$ \varepsilon_x : \mathscr H_x \to \mathscr H_{x, U} , $$ a notion I did not expect to exist. The relevant excerpt is below, with my thoughts at the end.
Here, $\mathscr C(\mathbb C^n)$ denotes the sheaf of germs of functions on $\mathbb C^n$ with values in $\mathbb C$.
Soit alors $x$ un point de $U$; on a un homomorphisme de restriction $$ \varepsilon_x : \mathscr C(\mathbb C^n)_x \to \mathscr C(U)_x . $$ L'image de $ \mathscr H_x $ par $ \varepsilon_x $ est un sous-anneau $ \mathscr H_{x, U}$ de $ \mathscr C(U)_x $; les $ \mathscr H_{x, U} $ forment un sous-faisceau $ \mathscr H_U $ de $ \mathscr C(U) $, que nous appellerons le faisceau des germes de fonctions holomorphes sur $U$; c'est un faisceau d'anneaux. Nous noterons $ \mathscr A_x(U) $ le noyau de $ \varepsilon_x : \mathscr H_x \to \mathscr H_{x, U} $; vu la définition de $ \mathscr H_{x, U} $, c'est l'ensemble des $ f \in \mathscr H_x $ dont la restriction à $U$ est nulle dans un voisinage de $x$; nous identifierons fréquemment $\mathscr H_{x, U}$ à l'anneau quotient $\mathscr H_x / \mathscr A_x(U)$.
An English translation I found online:
So let $x$ be a point in $U$; we have a restriction $$ \varepsilon_x : \mathscr C(\mathbb C^n)_x \to \mathscr C(U)_x . $$ The image of $\mathscr H_x$ under $\varepsilon_x$ is a subring $\mathscr H_{x, U}$ of $\mathscr C(U)_x$; the $\mathscr H_{x, U}$ form a subsheaf $\mathscr H_U$ of $\mathscr C(U)$, which we call the sheaf of germs of holomorphic functions on $U$; it is a sheaf of rings. We denote by $\mathscr A_x(U)$ the kernel of $\varepsilon_x : \mathscr H_x \to \mathscr H_{x, U}$; in light of the definition of $\mathscr H_{x, U}$, it is the set of $f \in \mathscr H_x$ whose restriction to $U$ is zero in a neighbourhood of $x$; we frequently identify $\mathscr H_{x, U}$ with the quotient ring $\mathscr H_x / \mathscr A_x(U)$.
I want to repackage this statement by applying the first isomorphism theorem to the exact sequence $$ \ker \varepsilon_x = \mathscr A_x(U) \to \mathscr H_x \to \mathscr H_{x, U} \to 0 $$ to get $\mathscr H_{x, U} \cong \mathscr H_x / \mathscr A_x(U)$, but I need to be precise about what exactly $\mathscr H_{x, U}$ and $\mathscr H_U$ denote. The restriction $$ \begin{align*} \mathscr H &\to \mathscr H_x \\ f &\mapsto f_x \end{align*} $$ makes sense, but what is $$ \begin{align*} \mathscr H_x &\to \mathscr H_{x, U} \\ f_x &\mapsto \ ? \end{align*} $$
The short answer is that $\newcommand{\C}{\mathbb{C}} \newcommand{\scrH}{\mathscr{H}} \mathscr H$ is the sheaf of holomorphic functions on $\mathbb{C}^n$, while $\mathscr H_{U}$ (which he is in the process of defining) is the sheaf of holomorphic functions on $U$. So $\mathscr H_x$ consists of germs of functions on $\mathbb{C}^n$, meaning equivalence classes $[f,V]$ of pairs $(f,V)$ where $V \subseteq \mathbb{C}^n$ is open with $x \in V$, and $f: V \to \mathbb{C}$ is holomorphic. As he has defined $\mathscr H_{x,U}$ to be the image $\varepsilon_x(\mathscr H_x)$ of $\scrH_x$ under $\varepsilon_x$, the map $\mathscr H_x \to \mathscr H_{x,U}$ is simply $\varepsilon_x$. More explicitly, given an equivalence class $[f,V] \in \scrH_x$, we have $\varepsilon_x([f,V]) = [f|_{V \cap U}, V \cap U]$.
An example might make this clearer. Suppose $n=2$, so $\mathscr H$ is the sheaf of holomorphic functions on $\mathbb{C}^2$, with coordinates $u,v$, and $U \subseteq \mathbb{C}^2$ is the $u$-axis, defined by $v=0$. Then the restriction map $\varepsilon_x$ is basically given by $f(u,v) \mapsto f(u,0)$. There are many holomorphic functions on $\mathbb{C}^2$ that restrict to the same holomorphic function on $U$. For instance, the maps $u$, $u + 37v - 51v^5$, and $u - 42 \sin(v)$ all restrict to same function on $U$.
For a more extreme example, one could even take $U$ to just be a point. Let $U \subseteq \C$ be the singleton containing the origin: $U = \{0\}$. Letting $x = 0$, then $\scrH_{x,U} \cong \C$: the only nonempty open subset of $U = \{0\}$ is $U$ itself, so the map $[f, U] \mapsto f(0)$ is an isomorphism. By contrast, I claim that $\scrH_x$ is not a field. For instance, the function $f(z) = z$ has a nonzero germ at $0$ (it takes on nonzero values on any open neighborhood in $\mathbb{C}$ containing $0$), but its germ $[f,V] \in \scrH_x$ has no multiplicative inverse. (If $[g, W]$ were such an inverse, then we would have $1 = f(0) \, g(0) = 0 \cdot g(0) = 0$, contradiction.) Thus $\scrH_x$ is not a field, hence in particular is not isomorphic to $\C \cong \scrH_{x,U}$.