I always had a peripheral understanding that there was something more to maths than working out how your change or divvying up sweets with your siblings. But the day I really, truly understood was when I learned about $\pi$.
$\pi$ was magical to me. For one thing, it's a funny-looking Greek letter with a funny-sounding name. But, more captivatingly, it introduced me to an epiphany: that somewhere, on some level, the fundamental structure of reality itself could be understood through mathematics.
Let's assume your childen understand what a circle is, and how to measure things with a measuring tape. Introduce them to circumference and diameter. Give them a table with three columns—circumference, diameter and "the secret of circles"—and a big tape measure. Tell them to go out and measure as many circles as they can find: plates, car tires, stop signs, plant pots, lines on a basketball court… anything so long as it's circular. Let them loose.
Later, once they're done measuring everything in the neighbourhood, hand them a calculator and tell them to go through each of their circles and divide the circumference by the diameter, and write the number they get in the mystery third column. Tell them that a pattern will start to appear, and they need to see if they can spot it.
Once they're done, you can explain to them that the reason the first couple of numbers is the same is because there's a number, a magical number, that tells us a secret about every circle in the universe—from rings we wear on our fingers to the sun and moon in the sky and the whole planet Earth. No matter how big or how small, how grand or how humble, every single circle is a bit more than three times bigger around than it is from one side to the other. This number is so special that it has its own name, pi, and its own special letter, $\pi$. It's not three and it's not four—it's somewhere just after three, and we can't write down exactly where because it goes on forever. Luckily, we only really need to know the first few numbers most of the time, so we can use this magical number whenever we need it.
The sense of revelation that came from knowing that every circle in the universe is connected by this strange, special number stayed with me for a long time, and is at least partly responsible for my love of mathematics in later life.