Let $p$ be a prime number. A global field is defined as a finite extension of $\mathbb Q$ or $\mathbb F_p(t)$. On the other hand one can show that a local field (which is by definition a complete discrete valuation field with finite residue field) is a finite extension of $\mathbb Q_p$ or $\mathbb F_p((t))$.
Why do we use the adjectives "local" and "global"? What is the geometric picture that I am missing?
In the early 1930s when local class field theory was developed, it was always called "class field theory in the small" (in German). Eventually the term local prevailed. In 1935, Hasse wrote, in a review of an article of Schilling, about the "local arithmetic structure of algebras". Later on, especially when articles in French and English on class field theory began appearing, the expression "local" was used, and the dichotomy small-large became local-global. In particular Chevalley's articles on local class field theory from 1938 use "theorie locale des corps de classes".
I don't think there's a big picture you're missing: in local fields, you always study one prime ideal; in global fields, you look at all of them at the same time. As a matter of fact, global fields can be characterized by a product formula (Artin-Whaples etc.).
Hopefully someone will correct me if I am wrong. In algebraic geometry over an algebraically closed field $k$, we have a notion of looking at things locally. This is done by looking at a stalk. If $X$ is an affine variety (by definition, an irreducible closed set in $n$-space $k^n$), and $R$ is the coordinate ring of $X$ (by definition, $k[T_1, ... , T_n]$ modulo the prime ideal corresponding to $X$), then we have a notion of a ring $\mathcal O_X(U)$ of regular functions for each open set $U$ of $X$. By definition, the elements of $\mathcal O_X(U)$ are functions from $U$ to $k$ which locally can be written as rational functions in $k$.
Now, if $x$ is a point of $X$, the stalk $\mathcal O_{X,x}$ is the ring of equivalence classes $(f,U)$, where $U$ is a neighborhood of $x$ and $f \in \mathcal O_X(U)$, where two classes $(f,U)$ and $(g,V)$ are the same if $f$ and $g$ agree on some neighborhood of $x$ which is contained in $U \cap V$. The upshot is that $\mathcal O_X(X)$ is canonically isomorphic to $R$, and if $\mathfrak m$ is the maximal ideal of $R$ corresponding to the point $x$, then $\mathcal O_{X,x}$ is canonically isomorphic to $R_{\mathfrak m}$. This is where 'localization' gets its name, because from the geometric side we are literally looking at the local property of functions.
Instead of looking at a variety $X$, you can create similar parallels by looking at an arbitrary commutative ring $B$ and its associated 'affine scheme,' but I don't think these notions were popular until Grothendieck.
Now if $K$ is a number field with ring of integers $A$, then many results about the structure of $A$ are determined by looking at its prime ideals, which can be isolated by localizing. Now if $P$ is a prime ideal of $A$, and $v$ is the absolute value corresponding to $P$, then many properties about $A$ are connected to the completed ring $A_v$.
Since $P$ completes to the unique maximal ideal of $A_v$, it is natural to refer to the quotient field of $A_v$ as a local field, because its properties carry over to properties about the localized ring $A_P$.
In addition to the other answers one could give a number-theoretic "picture", e.g., the Hasse–Minkowski theorem: A quadratic form over a number field nontrivially represents zero if and only if this holds for all completions of the field. So we have a nontrivial solution, say, over the global field $\mathbb{Q}$ if and only we have a nontrivial solution over all local fields $\mathbb{Q}_p$, $p$ prime and $p=\infty$. This illustrates the meaning of "local" and "global" with respect to solutions of polynomial equations ( see also the "local-global" principle in general, also discussed here). This may be more concrete than a "geometric picture".
My path to understand this is:
The reason that those $\mathbb{Q},\ \mathbb{R}$ and $\mathbb{C}$ showing up in the end is that we only consider a field with an absolute value, which is a function $|\cdot|:K\to \mathbb{R}_{\geq 0}$ with three properties. This is where $\mathbb{R}$ appears in the axioms.
For a general valued field, its value group is a totally ordered abelian group, e.g. $\mathbb{R}_{>0}$ with multiplication. And $\mathbb{Z}$ with addition is the simplest totally ordered abelian group so discrete valuation ring has its own name.
Moreover, we can see that in the definition of valuation it's always non-Archimedean. In fact every valuation with value group $\mathbb{R}_{>0}$ gives a non-Archimedean absolute value, and the converse is also true. So valued field is not quite the same as a field with an absolute value, there is no "Archimedean" valued field.
Note: You may find that in the axiom 2 of Artin-Whaples' paper it mentions $\mathbb{R}$ and $\mathbb{C}$ (and you may forget to look at the footnote like I did), but they can be dropped by Ostrowski's theorem: Let $F$ be a field complete w.r.t. an Archimedean absolute value $\alpha$, then $F$ is isomorphic to either $\mathbb{R}$ or $\mathbb{C}$, with their usual Archimedean absolute value.
