Firefox has in-built spell check, but by default, this spell check is available only in Text area (usually multiple lines with scroll bar) entry fields. Text entry boxes (usually 1 line) will not show spelling mistakes. To activate spell check in text fields, open the configuration page of Firefox. Type about:config in address bar and press enter. In the filter text-box, type: layout.spellcheckDefault
The default value of this preference is 1. Double click on this parameter and change the value to 2. Close configuration page and now open any page which has a text-box and check for wrong spelling. In case the spell check is still not enabled, right click on the text-box to pop-up the menu and check the Check Spelling menu item.