There are two orderings of the set $\mathbb N = \{0,1,\dots\}$:
- magnitude $a \leq b$
- divisibility $a\mid b$ (i.e. $\exists c. b = a c$)
They are mostly compatible - usually when $a \mid b$, it holds $a \leq b$.
Some definitions are phrased using "greater than" ordering, while in fact the "divisibility" ordering is the real essence.
For example, the greatest common divisor of $a$ and $b$ might be defined as the greatest number which is a common divisor of both $a$ and $b$. Characteristic of a ring $R$ might be defined as smallest number $n>0$ which satisfies $n \cdot 1 = 0$.
Under such commonly taught definitions, it seems natural that $\operatorname{gcd}(0,0)=\infty$ and $\operatorname{char} \mathbb Z = \infty$.
However, those definitions implicitly rely on ideals, and are better phrased using divisibility order. The incompatibility is then more visible: $0$ is the largest element in divisibility order, while it is smallest in magnitude order. Magnitude has no largest element, and often $\infty$ is added to cover this case.
So let's formulate the definitions again, but this time using divisibility ordering.
- The greatest common divisor of two numbers $a,b$ is greatest number (in sense of $\mid$) that is a divisor of $a$ and $b$ (i.e. is smaller than $a$ and $b$ in divisibility ordering). This is prettier - $\operatorname{gcd}$ is now the $\wedge$ operator in lattice $(\mathbb N, \mid)$; it also forms a monoid, with $0$ as identity element. Additionally, the definition can be adapted to any ring.
- The characteristic of a ring $R$ is the smallest number $n$ (in sense of $\mid$) that satisfies $n \cdot 1 =0$. As a bonus, compared to previous definition, we can remove the $n>0$ restriction: zero is always a valid "annihilator" but it is often not the smallest one. Now we get $\operatorname{char} \mathbb Z = 0$.
Characteristic is a "multiplicative" notion, like gcd. If you have a homomorphism of rings $f: A \to B$, it must hold $\operatorname{char} B \mid \operatorname{char} A$. For example, you cannot map ${\mathbb Z}_2$ to ${\mathbb Z}_4$ - in a sense, ${\mathbb Z}_2$ is "smaller" than ${\mathbb Z}_4$. "Bigger" rings have "more divisible" characteristic, their characteristics are greater in the sense of divisibility. And the "most divisible" number is 0. Another example is $\operatorname{char} A \times B = \operatorname{lcm}(\operatorname{char} A, \operatorname{char} B)$.
In a bit more abstract language: given any ideal $I \subseteq \mathbb Z$, we associate to it the smallest nonnegative element, under the divisibility order. By properties of $\mathbb Z$, every other element of $I$ is a multiple of it. Let's call this number $\operatorname{min}(I)$.
We can now define $\operatorname{gcd}(a,b)=\operatorname{min} ((a) + (b))$, and $\operatorname{char} R = \min (\ker f)$, where $f \colon \mathbb Z \to R$ is the canonical map.
The definition of $\operatorname{min}(I)$ works for any PID, it does not require magnitude order. In any PID, $I = (\operatorname{min}(I))$.
(I dislike saying the ideal $\{0\}$ is "generated" by $0$; although this is true, it also generated by empty set. We do not say that $(2)$ is generated by $0$ and $2$.)
Best Answer
It is simply a statement about the maximum additive order of something in the ring.
Quotients of $\Bbb Z$ are a natural source of rings with different characteristics, of course.
The most physical analogy that comes to mind is modular arithmetic. If you're familiar with any sort of cyclic behavior that repeats after finitely many steps, you can view the characteristic of the ring as a "period" of the cyclic behavior.
The choice of $0$ to represent the case when there is no finite period is purely a conventional one. See also Why “characteristic zero” and not “infinite characteristic”?