Asides are generally short posts or notes — the total opposite of a normal blog post. The post formats section of the Codex describes an aside as:
Typically styled without a title. Similar to a Facebook note update.
I think of them as something aside from my regular flow of content, a relaxing break from all the work that goes into producing a quality, full-length article. Some bloggers use the aside as an all-encompassing format that includes aside, link, quote, and status posts.