You are using Firefox version 35.0 but the latest realeased version is 37. So an window is appearing to ask you to install updates. However, you can choose update later.

If there are problems with updating or with the permissions then best is to download the full version and trash the currently installed version to do a clean install of the new version.

Download a new copy of the Firefox application and save the disk image file to the desktop

• Firefox 37.0.1: https://www.mozilla.org/en-US/firefox/all/

• Trash the current Firefox application (open the Applications folder in the Finder and drag the Firefox application to the Trash) to do a clean (re)install
• Install the new version that you have downloaded
• https://support.mozilla.org/kb/Installing+Firefox+on+Mac

Your personal data is stored elsewhere in the Firefox profile folder, so you won't lose your bookmarks and other personal data when you uninstall and (re)install or update Firefox.

• http://kb.mozillazine.org/Profile_folder_-_Firefox