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 300x300, 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="<?php echo get_attachment_link( get_post_thumbnail_id() ); ?>">
<?php the_post_thumbnail( 'single-post-image' ); ?>
</a>
<?php } ?>