In order to "Eradicate" a disease, you must first cure that disease, and then remove all cubes of its color off the board. Once eradicated, there should be 24 disease cubes in the supply (shown on the top of the screen).
The easiest and fastest way to eradicate a disease once you've cured it is to send the Medic through that colored region. Their ability lets them automatically treat cubes from cities they are in without taking an action. With 4 movement actions, the Medic can clear 4 cities of cubes.