Autoionization and the reverse reaction are constantly happening in water, and when the reaction is happening at the same rate forward and backward the system is said to be "at dynamic equilibrium" (aka, stuff is happening, but there's no net change)

In pure water, the equilibrium concentration of hydronium and hydroxide are equal, so it's said to be neutral. At room temperature, that equilibrium concentration is approximately 1*10^-7 moles per liter, which gives a pH of 7 (since pH is defined as the negative log _10 of hydronium concentration)

I once brought up in a family dinner how incredible and strange water is, and how we don't really think about it.

It appears naturally in all three phases, expands when frozen, has a high surface tension, has a high specific heat, and can behave as a mild base or acid. Oh, and all the living stuff has water in it.

Nobody really understood what I meant except my sister.

I know it's a joke and all, but it's not just the pH that makes something burn. A regular coke has a pH of around 2.5, for example.

In high school I was told that one in avagadros number of water molecules splits into ions.
Is that right? It seems like a very small amount.

Well, yes and no. The pH scale follows the hydrogen ion concentration, but specifically in aqueous media. The reason 7 is in the "middle" of the scale is because the natural dissociation of water sits at equilibrium at 10^-7 M H+ at 298K, IIRC. So perturbations naturally just displace that specific equilibrium, so it absolutely is normative to water.