When people talk about the eradication of something, they are referring to its total destruction. Imagine a movie villain destroying a planet with a massive laser beam and you'll have the right idea.
If a gardener's goal is the eradication of every single weed in his back yard, he might start by uprooting dandelions. Fittingly, the root of eradication is the Latin word for "uproot," ērādīcāre. The eradication of anything, whether it's disease, poverty, war, or weeds, involves removing every last trace of it, right down to the roots.