Widgets are used to add features and functionality to the sidebar (usually) of your blog. Widget placement is dependent on themes.
If you change themes your widgets and settings might not be preserved. You can drag your widgets to the Inactive Widgets section of the widgets page in your dashboard to preserve their settings.
This blog has just been created with a standard set of widgets.
You can access the widget dashboard from the Appearance section. You may have to show that section using the drop down disclosure arrow.
On the Widget page of your dashboard you add widgets to the sidebar by dragging them.
After adding widgets many need to be configured. For example adding a text or about widget will need some text filled in. The Widget will need to be Saved  too.
The above widget on the side bar. We have also added a Tag Cloud Widget.
Pages about specific widgets: