What does that mean?

Well, it means different things for different apps. Whenever you connect your Facebook account to a mobile app or website, there are a list of other things the app is asking for permission to access. For example, access to post to your wall or view your tagged photos. When you allow any Facebook application to access your account, you are giving them temporary (up to about 15 minutes) to access and use your data. For example, if I grant an application access to my Facebook wall data today, they can’t see it tomorrow. Unless… the app specified that they get access to my data at any time. In that case, they can see all my wall posts tomorrow, too. Without my consent then.

For the developer and whoever had the app made, they get a oAuth token that allows them to act as you on Facebook given the parameters of whatever you gave them access to. Their app requests a thing called “offline_access” in their code… which is exactly what it means. Access to your account when you’re not  online.

Oh no! You didn’t mean to grant an application that permission? Well, this is a great post on how to stop any app… 

The Offline Access permission is typically only a good if the app you’re using needs to update your Facebook wall status or something else automatically when certain events occur and you don’t want to have to log into Facebook all the time to do those things.