Hi,
1. Sections are the top level of the hierarchy and categories are the second. So you can define sections and in each section you can define categories. This changes in Joomla 1.6 which is currently in beta - here you only have categories but they can be nested to any depth.
2. Extension is the generic name for anything added to the Joomla core. An extension generally comprises one or more of the three types you mention. A plugin is a behind the scenes service that operates in the background (such as indexing a site for a search function). A module defines the display of something in a template position that is an optional addition to a page (such as a search box). A component defines the main display on the page (such as a search results page). These types apply to the core as well. The display of any of the standard content pages is controlled by com_content for example. Menus are common examples of modules. Caching is handled by the System-Cache plugin.
3. The core Login module has the option to show a greeting when a user is logged in - normally Hi yarekGmail if that is the username. I have not done it but I expect you could check the code in the login module to use this greeting in articles as well if you wished.
Cheers - David
