"Petrichor" is the wonderful word that describes the wonderful scent of the air after a rain shower. It comes, like so many wonderful words do, from the ancient Greek: a combination of ichor, the "ethereal essence" the Greeks believed flowed through the veins of their gods, and petros, the stones that form the surface of the Earth.
In the video above, PBS's Joe Hanson describes the biology that leads to petrichor. "When decomposed organic material is blown airborne from dry soil," he explains, "it lands on dirt and rock where it's joined by minerals. And the whole mixture is cooked in this magical medley of molecules. Falling raindrops then send those chemicals airborne, right into your nostalgic nostrils."
When it's not raining, though, that molecular mixture serves a different purpose: signaling plants to keep their roots from growing and their seeds from sprouting. No use wasting energy on all that, after all, when there's no water to be drunk. (Or, as Hansen puts it, "Petrichor: it's for the plant that's tired of waiting … to germinate.")
So why does the world smell different to humans after it rains? Because of plants, basically.