hi harah, if this is an error message shown within the skype application, it will be unrelated to firefox and offtopic in this forum, since we are not able to provide help for third-party products. please contact skype's support instead: https://support.skype.com

Isn't my browser (Firefox) responsible for cookies? Why does "please enable cookies" message come up in Facebook/Skype even though I have cleaned the cache and left everything open?

You can inspect and manage the cookies for the domain in the currently selected tab:

Click the "Site Identity Button" (globe/padlock) on the location/address bar
Click "More Information" to open "Tools > Page Info" with the Security tab selected
Click "View Cookies" to view the cookies for the domain in the currently selected tab

Check the cookie permissions on the Permissions tab of Page Info.

cookie settings in firefox won't have an effect on separate applications like skype - it may be that they depend on IE settings (since they are also part of microsoft) as they use it to generate the webview for the login screen . not sure about that though...

In case Skype is using Flash storage instead of Firefox storage, can you check to make sure you are allowing at least 10KB of storage here (also see screenshot):

http://www.macromedia.com/support/documentation/en/flashplayer/help/settings_manager03.html