Featured image thumbnail sometimes in content, sometimes changes header images

Assuming you are using Twenty Ten, Twenty Eleven, or one of the several Themes that derive Post Thumbnail (i.e. “featured image”) feature handling from either of these Themes:

• The Featured Image is applied to the header image, if and only if the dimensions of the Featured Image exceed the dimensions of the header image.
• The Featured Image is not displayed in the Post Content by default; you must manually insert the Featured Image into the Post Content, as you would any other image, if you want it to appear in your Post.

EDIT

To have a custom image size, such as 300×300, you can use add_image_size(). For example, add the following to functions.php:

add_image_size( 'single-post-image', 300, 300, true );

Then, in single.php, you can add:

<?php the_post_thumbnail( 'single-post-image' ); ?>

And your 300x300px image will be inserted wherever you place it.

(Note: you may need to regenerate existing Thumbnails.)

EDIT

To link your displayed, custom-sized featured image to its attachment-page view, you need to use get_attachment_link(). In single.php:

<?php if ( has_post_thumbnail() ) { ?>
    <a href="https://wordpress.stackexchange.com/questions/23443/<?php echo get_attachment_link( get_post_thumbnail_id() ); ?>">
    <?php the_post_thumbnail( 'single-post-image' ); ?>
    </a>
<?php } ?>