I want to understand this classic example of sheaves sequence
$$ 1\to \mathbb{Z}\to \mathcal{O}_\mathbb{C} \to \mathcal{O}_\mathbb{C^*} \to 1$$
$ \cdot \mathcal{O}_\mathbb{C}$ is the sheaf of the holomorphic functions on $\mathbb{C}$
$\cdot \mathcal{O}_\mathbb{C^*} $ is the sheaf of the holomorphic functions on $\mathbb{C}\setminus \{0\}$
I can understand that exp function gives us $\mathcal{O}_\mathbb{C} \to \mathcal{O}_\mathbb{C^*}$.
The questions I have regarding the sequence are:
What is $1$ stands for ? And how we can get $ 1\to \mathbb{Z}$,
$\mathbb{Z}\to \mathcal{O}_\mathbb{C}$ and $\mathcal{O}_\mathbb{C^*} \to 1$ ?
Why this sequence is important ? What information we get from this ?
As Kenta S points out in the comments, the sequence might be better written as $$ 0 \to \underline{\Bbb{Z}} \to \mathcal{O}_{\Bbb{C}} \to \mathcal{O}_{\Bbb{C}}^*\to 0 $$ where the map $0\to \underline{\Bbb{Z}}$ is the inclusion of $0$ as the constant function $0(z) = 0$. Also note that $\underline{\Bbb{Z}}$ denotes the constant sheaf with values in $\Bbb{Z}$. Over a connected open set $U$, $$\underline{\Bbb{Z}}(U) = \{f:U\to \Bbb{C}: f(z) = n\in \Bbb{Z} \:\text{for all}\: z\in U\}.$$ Viewing a constant function as a holomorphic function gives a natural inclusion $\underline{\Bbb{Z}}\to \mathcal{O}_{\Bbb{C}}$. The map $\mathcal{O}_{\Bbb{C}}^*\to 0$ is the map sending every function to $0$. Such a map always exists. In what follows, I will replace $\Bbb{C}$ by an arbitrary complex manifold $(X,\mathcal{O}_X)$, which is essentially a choice of a topological space with a sheaf of functions valued in $\Bbb{C}$ called "holomorphic" (although the existence of such a structure is a rather special situation). For concreteness, you could just take $X = \Bbb{C}$, although this sequence will be more useful in cases such as $X$ being a Riemann surface. Associated to a short exact sequence of sheaves $$ 0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0 $$ is a long exact sequence of sheaf cohomology groups: $$ 0\to H^0(X,\mathcal{F})\to H^0(X,\mathcal{G})\to H^0(X,\mathcal{H})\to H^1(X,\mathcal{F})\to \cdots\to H^n(X,\mathcal{F}) \to H^n(X,\mathcal{G})\to H^n(X,\mathcal{H})\to 0. $$ Here, $n$ is the dimension of $X$. In this particular case, the short exact sequence relates three very important series of sheaf cohomologies. It turns out that $H^i(X,\underline{\Bbb{Z}}) = H^i_{\mathrm{sing}}(X;\Bbb{Z})$, the singular cohomology of $X$. These groups are pieces of topological data. On the other hand, $H^i(X,\mathcal{O}_X)$ packages information about the complex structure on $X$, as does $H^i(X,\mathcal{O}_X^*)$. Notably, $H^1(X,\mathcal{O}_X^*) \cong \mathrm{Pic}(X)$, which is the group of holomorphic line bundles on $X$. These are important when defining embeddings of (compact) manifolds into projective spaces. Anyway, to see this principle in action we can write out the first few rounds of the sequence: $$ 0\to H^0(X,\Bbb{Z})\to H^0(X,\mathcal{O}_X)\to H^0(X,\mathcal{O}_X^*)\to H^1(X,\Bbb{Z}) \to H^1(X,\mathcal{O}_X)\to \mathrm{Pic}(X)\to H^2(X,\Bbb{Z})\to\cdots $$ The map $\mathrm{Pic}(X)\to H^2(X,\Bbb{Z})$ gives a way to define the first Chern class map, $c_1:\mathrm{Pic}(X)\to H^2(X,\Bbb{Z})$ which is important in the theory of characteristic classes (and in many other places including intersection thory).
In one line: this sequence relates the complex structure of the space $X$ to the topological invariants of the underlying topological space $X$.