Characteristic of a ring: intuitive explanation

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”?

I agree with the above answer completely and would like to add little bit more to it.

Most familiar rings to us are $\mathbb{Z}, \mathbb{Q}$ and $\mathbb{Z}/n\mathbb{Z}$ and once derived from them (function rings, polynomial rings, matrix rings etc). It's not difficult to see that all of these are somehow connected to (or obtain from) $\mathbb{Z}.$ We can make this comparison mathematically precise by observing that for any ring $R$ (with unity) there is a canonical ring homomorphism $\varphi : \mathbb{Z}\to R.$

The kernel of this map (the data that we lost during the transformation) is an ideal of (the principal ideal domain) $\mathbb{Z}.$ Therefore we have a smallest (non-negative) integer that generates it, say $m$. This is exactly the characteristic of $R.$

Moreover, by the first isomorphism theorem we have that $$\mathbb{Z}/m\mathbb{Z}\cong \operatorname{im}\varphi,$$ which is a sub-ring of $R.$ Also since $$m.1_R=0$$ the ring $R$ become a $\mathbb{Z}/m\mathbb{Z}$-algebra. An instructive example for this is Boolean rings. Furthermore, if $R$ is an integral domain then $m$ must be a prime number or zero. So, this number $m$ (characteristic) carries some important information about the ring $R.$