Why does the punctured affine line deserve its name? Let $k$ be a (commutative unitary) ring. The subobject $\operatorname{Spec} k[x,x^{-1}]\rightarrowtail \operatorname{Spec} k[x]=\mathbb A_k ^1$, where $k[x,x^{-1}]=k[x,y]/ \left\langle xy-1 \right\rangle$, is often called the punctured affine line. I've even seen $\operatorname{Spec} k[x,x^{-1}]$ denoted $\mathbb A_k^1\setminus \left\{ 0 \right\}$.
Why does $\operatorname{Spec} k[x,x^{-1}]$ deserve this name and notation? Isn't this just the hyperbola $xy=1$ in the affine plane $\mathbb A_k^2$? On the other hand, if we look at $k[x,x^{-1}]$ as the localization of $k[x]$ at $ \left\{ x,x^2,\dots \right\}$, then $\operatorname{Spec} k[x]_x$ does look like we're looking at the points in which the polynomial $x\in k[x]$ is nonzero, which is "everything but zero"...
 A: Note that $\mathbb A_k^1\setminus \left\{ 0 \right\}$ is a Zariski open subset of $\mathbb A_k^1$ (being the complement of the zero set of $x$), and the projection $\operatorname{Spec} k[x,y]/(xy-1) \to \mathbb A_k^1\setminus \left\{ 0 \right\}$  is an isomorphism of locally ringed spaces (the latter with structure sheaf being a restriction of the structure sheaf of $\mathbb{A}_k^1$).
A: The details you seek can be found in Hartshorne. The claim is that the projection map from $X = V(xy-1) \subset \mathbb A^2$ to $Y = \mathbb A^1 - 0$ is an isomorphism of ringed spaces (I am thinking of $X$ and $Y$ as schemes). This means two things. The first thing is that the projection map is a homeomorphism, which I leave you to verify. $\mathbb A^1-0$ has the subspace topology from the Zariski topology from $\mathbb A^1$. The second is that the induced map $f^\#: \mathcal O_{Y} \to f_*\mathcal O_{X}$ of sheaves is an isomorphism (Hartshorne page 72-73). To verify that this map of sheaves is an isomorphism, one can check that the induced maps across stalks are all isomorphisms (page 63, Prop 1.1).
It is probably easier and better to first learn what a variety is, and to do this in the category of varieties first. The fact that we call the hyperbola the punctured affine line is a geometric idea (at least to me)
