Zombie Villagers are a more recent addition to Minecraft. Essentially, the villagers get hit with a disease that turns them into zombies. However, you can cure them and activate their trading post.
First, you have to inflict the Weakness status on the Zombie, which you can do with a potion. After that, you give it a Golden Apple. After around 2-5 minutes of waiting, the Zombie Villager should now be cured.