Here’s a post originally created using the “Aside” format under WordPress 3.4.2 with the default Twenty Eleven theme.  Note the more abbreviated listing for this type entry in the main blog page.  Here are some differences from the Standard format in the way that the Aside format posting is listed in the main blog page and various archive pages:

• the entry just has “ASIDE” as its heading, instead of the actual post title
• there is no comments bubble on the post heading line
• the original posting date is shown at the bottom of the post, instead of the top
• there is no indication of category or tag assignments for this post
• the only link to the separate post page is the one attached to the posting date

The corresponding post page does display the post title, categories, tags, and author, in the same way as a Standard format posting does.  Note, however, that the page for this individual posting does not provide any indication that it is an “Aside” format post.

What http://codex.wordpress.org/Post_Formats says about this format:

aside – Typically styled without a title. Similar to a Facebook note update.