What happens depends on the options the user selected when they chose to ignore you. At its most basic level, ignoring a user hides their posts from view--all their posts, past and future. The post is replaced with a link that lets you view the post if you want to. However, if someone else quotes them, that quote is visible to you. There's no easy way of blocking that.

You also have the option of preventing them from sending you private messages, preventing them from viewing your profile, and preventing them from sending you e-mail via the user contact form (which isn't implemented yet).