Question: When a client edits a featured image, they notice that they have the ability to set its "Alignment". If they were working with standard images embedded via the Editor, then this wouldn't be a problem. But since it is a featured image, WordPress doesn't pass in the appropriate class.

Is there a way to know which alignment option was set? If so, I can use that to add a class to the featured thumbnail accordingly.

Answer: 
Short answer: **you don't**.

The alignment settings only apply to images that the user *adds to the Post Content*. The Featured Image is *incorporated into the template* and is *not added by the user* to the Post Content. 

The Featured Image is directly integrated into the template, usually as a "post thumbnail" displayed on archive index pages, or as a "post header image" displayed above the post on archive index pages and/or single blog posts. In such cases, the Theme developer accounts for the *presentation* of the Featured Image via the template markup and CSS.

(That's not to say that the user can't manually add to the post the image used as the Featured Image; the user certainly *can* - and in which case, the manually added image will use/apply the alignment setting.)
As a tag-along to Chip's answer (which is correct), here's a sample of the meta that's stored for image attachments via `wp_get_attachment_meta($attachment_id)`:

```
array (
  'width' => '228',
  'height' => '147',
  'hwstring_small' => 'height='82' width='128'',
  'file' => '2009/06/1.jpg',
  'sizes' => 
  array (
    'thumbnail' => 
    array (
      'file' => '1-150x147.jpg',
      'width' => '150',
      'height' => '147',
    ),
  ),
  'image_meta' => 
  array (
    'aperture' => '0',
    'credit' => '',
    'camera' => '',
    'caption' => '',
    'created_timestamp' => '0',
    'copyright' => '',
    'focal_length' => '0',
    'iso' => '0',
    'shutter_speed' => '0',
    'title' => '',
  ),
)

```

The image description is stored in the attachment `post_content` column in the `posts` table.