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As is typical in Drupal, there's more than one way to get into a block's configuration panel.

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Blocks - Configurations - Block Manager Screen

1. Locating Block Configurations Via the Admin Menu

A permitted user will always be able to go to Admin > Structure > Blocks to access the block management screen.

Here, to access the configuration screen for a specific block, just click the configure link for the block in question.

Blocks - Configurations - Contextual Link

2. Locating Block Configurations Via a Contextual Link

Hovering over a block will trigger a contextual menu to to appear in the upper right corner of the block you're hovering on. This menu will appear as a little, clickable gear. When you click on it, it will expand and give you, at a minimum, a menu item that says Configure Block. Clicking on the Configure Block menu item will open that block's configuration settings.

Caveat

What if you want to show the same block in different regions on different pages, though? Is this possible?

No, it's not possible. Not using the configurations available here, at least.

This situation is really common in big sites that have multiple sections that are laid out in different ways, or in sites that use Organic Groups. This is where we need to move to a more robust tool to do our block configuration and layout - the Context module.