A Wordpress Widget is similar to a Wordpress Plugin in the way that a widget can add additional functions and features to your site.
In most cases, the most common area within your Wordpress theme to be Widgetized is usually the sidebar, but depending on your theme, you can often have Widgetized areas in almost all areas of your theme.
To give you an example of a theme with multiple widget areas the following screen shot is taken of the Hybrid Theme from Justin Tadlock.

Usually by default, your theme has certain elements coded into your sidebar, It’s usually your Blogroll, Meta area, Categories and Pages. Widgets allow you to change these default settings without having to mess with the code within your themes .php files.
For example if I only wanted to display my categories and popular posts without listing any pages or links to my Admin area in my sidebar, all I would need to do is add the Categories and Popular Posts widgets into my sidebar widget area.
To access your widgets, log into your Wordpress admin, and click on the Widgets link under the Appearance section on the left sidebar within your Wordpress admin.

After clicking on the Widgets link, your widgets admin area will open up, this will be where you see all of your Available Widgets and the Widgetized Areas for your theme.

- Available Widgets
- Widget Areas within Your Theme.
To activate an Available Widget all you do is drag the particular widget from the Available Widgets section into the specific Widgetized Area where you want the widget to be visible on your theme.
Sometimes additional settings are needed to set up and customize your widgets even further, save all changes and your done.
Wordpress usually comes pre-installed with some widgets, but if you require additional widgets for your site you can search the Wordpress Codex for more available widgets.
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Hi Jeremy, still not sure what a widget does though. What additional function to my website would I need?
Hi Kurt,
The additional functions you want for your site are entirely up to you.
But using Widgets just allows you to manipulate and change the functions on your site rather than having to hard code your .php files.
If you look at the example in the post, where I mention wanting to display your categories and popular posts, rather than the default settings in your Wordpress install, instead of removing the code from your sidebar.php file you simply add the correct “Widget” into your Sidebar “Widget Area”
Hope this helps.