WordPress Plugins and Widgets are specialized code modules used to extend out the functionality of a WordPress web site.  Examples of Plugins are the Aksimet AntiSpam Plugin – which helps screen visitor comments and determine if they were written by a human or a machine (spam), the Microkid Related Posts Plugin – which allows you to suggest similar Posts at the bottom of a current Post to the reader, and the ScribeSEO Plugin – which tests a Post for its’ Search Engine readiness, VaultPress Plugin – a WordPress site and database backup tool, and the list goes on…

A WordPress Widget is a specific type of WordPress Plugin that is used to add visual appeal and interactive options to a WordPress web site. Widgets were originally confined to the Sidebar of a WordPress Theme but can now be used almost anywhere including the header and footer of a Theme. Examples of Widgets include tag clouds, search boxes, calendars, weather updates, and surveys.

If you look on the right hand side of this web page, my Twitter feed, called “The Tweetisphere”, is an example of a theme widget as is the list of “Categories” for my JasonPalmer.com site.  And, if you look at the bottom of this page, you will see the Related Posts Widget, with the title “Other related stories…”

Plugins and Widgets are created by army of independent software developers in the WordPress Community. (Check out the WordPress Plug-in Listings here.)

Many users will only need just a few Plugins and WordPress includes some popular ones, such as Akismet, by default.  However, there are thousands to choose from to do things that previously required the expertise of a programmer to create.

Many Commercial WordPress Theme Developers such as StudioPress.com, the creators of the Genesis Framework, make popular plugins and widgets, such as the “Social Profiles Widget” – used to insert icons links on a site page to your profiles on popular social sharing sites such as Twitter, Facebook, Linked In, FeedBurner, and more, which are optimized and tested specifically for their Genesis Themes.  As the WordPress Theme Widget specification is a standard, most Widgets, even if developed by a specific Theme creator will work with other WordPress themes without issue.

Plugins (and the Widgets they create) are installed in WordPress through the Plugin Manager of the WordPress Administrative Dashboard. You can usually search by name or keyword and then install the Plugin by clicking on “install.”  Once installed, a Plugin needs to be activated, and in some cases, options configured.

In the case of some Commercial “Paid” Plugins, a special compressed file in .ZIP format is provided by the commercial developer and then you, as Administrator, through the WordPress Plugin Manager, upload the .ZIP file to your site, then activate it, and configure options as necessary.

WordPress.org keeps track of over 20,000 Plugins and Widgets and will tell you which are the most popular, in terms of number of times downloaded by users as well as ratings as voted on by the WordPress community.

Check out the WordPress Plugin Listing:
http://wordpress.org/extend/plugins