A featured image is an image that can be used to be displayed with your post in your blog’s homepage or in some custom theme pages (examples: portfolios, archives, etc, depending on the theme).
Most themes use an option in the post editing page called Featured Image.
Click on the link displayed and choose an image from your computer or one already uploaded to your Media Library.
A thumbnail of the image will then appear in that box:
Basically, it’s this simple, but here are some things you should have in mind:
This image may or may not be included in the post itself – you might want to only show the image in the homepage and not in the single post page.
If the image is showed twice (in the homepage and in the single post page) and you did not insert it in your post, it’s because that option is active in your theme. Please go to Appearance → Theme Options and disable the option Show the attached image in the single post page.
In some themes, you may not need to define a featured image. Check in your theme’s options (Appearance → Theme Options) if there’s any available option that will use the first uploaded image in the post as the featured image.