Personally, I don’t use custom post formats too much. These are not custom post types, but rather a way of identifying whether a post is an article, quote, link, etc. If you’ve used Tumblr, you know what I’m talking about.
On the Make WordPress blog, the development team commented that post formats are one of those features that have a lot of potential, but just haven’t really been an intuitive and focused part of the admin UI. This new release will change that. They said the design will do something like what the WordPress Post Formats Admin UI plugin does. Here are some examples of what that plugin looks like:



One of the reason’s that post formats don’t get used as much is partially because not too many themes really leverage them (especially not the default WordPress themes). The new Twenty Thirteen theme changes this though and has a big focus on post formats on the front end.
I think that we will very quickly see more emphasis on post formats coming from theme developers after 3.6 is released.
I’m most curious how the new UI will handle the addition of custom post formats. On a very related note, I have planned to do some custom post formats on WP Zac in the near future.