You’re not looking at “different types of posts,” you’re looking at post formats. The format types are standardized in core (i.e. video, photo, default), but the presentation of these formats is 100% up to the theme.
That is, there is no “WordPress-approved manner” or standard for presenting previews of post formats.
There are, however, great examples of post formats in the wild:
- Twenty Eleven, which ships with WordPress
- Tapestry by StudioPress
- Twenty Twelve the new theme coming out with WP 3.5
If you need specific examples of how to implement differing presentations in your own theme, please feel free to ask a new question.