Warning: This description contains spoilers about the pilot.
In Jeremiah, a plague has swept the planet, wiping out virtually everyone above puberty. Naturally, civilization collapsed.
Cut to some 10(?) years later, where the survivors have grown up. It's a world without most of the things we take for granted, like electricity, TV and movies, underwear, and so forth. Guns and cars are still around, but rare, as gas and bullets are valuable commodities.
The star is Jeremiah, played by Luke Perry (who'd've thunk he could really act?), and his best friend and partner is Kurtie, played by Malcolm Jamal Warner. Jeremiah is a really altruistic guy, and one of the main themes in the show is making the world a better place. It's also a buddy show, as the two main characters have a pretty good relationship.
There are also a few quests in the show. Jeremiah is searching for "Valhalla Sector," which is where his parents were heading before he got separated from them, just prior to the Big Death. There's another major plotline tying into many episodes of season 1, but I'll stop there.
In the pilot, Jeremiah meets Kurtie and they form something of an alliance. Jeremiah crosses the path of a guy who's from "The End of the World," a place of legend where they have electricity, gas, and so on, except it's a real place. The End of the World turns out to be the NORAD command base ("Thunder Mountain"), designed to survive just about anything. The survivors of Thunder Mountain have been waiting, trying to figure out how they can best help the world, without the world finding out about them and coming in to take everything. Jeremiah and Kurtie become field agents for them, going on various missions related to the ultimate goals of the people at Thunder Mountain.
It's a good show. It's not really science fiction, although it is post-apocalypse. Like Babylon 5, there's clearly a grand story arc that's behind most of the individual episodes. It's arguably the best thing J. Michael Straczynski has done since Babylon 5.