If you click on the "Styles" control in TextEdit's toolbar, you will at first only see Favorite Styles.

A Favorite Style can be used in any TextEdit document for this user on this computer, but nowhere else. It is stored in the hidden file .GlobalPreferences.plist.
If you click on "Other...", you will get to a window where you can see the Document Styles. In contrast to almost all other programs, a Document Style in TextEdit has no customisable name.

It just contains some sample text ("ONE" in the picture above), which has a particular format. All formats used in the document appear as you click the arrows back and forth.
To create a Favorite Style from the current format, click on the button "Add To Favorites" and give it a name, and choose if you want the font and ruler information to be part of the style.
To delete a Favorite Style, click on the radio button Favorite Styles, select the style you want to delete and click on Remove From Favorites.

If you click on Select, you can choose to select where the style is used, based on different criteria.

You click on Apply, the active style is applied to the currently selected text.
Done in the Style window just closes the window.
If you do not have the Ruler active, you can still use Styles with the menus: Format > Font > Style
There is no easy way to edit an existing Favorite Style. The steps are:
- Apply the style to some text.
- Modify the text to the new format you want.
- Add the new format as a new Favorite Style with the same name as the old style.
- Accept the warning that you are replacing an existing style.
Personally, I think it is questionable if TextEdit's styles are better than no styles at all. It is clear that there has been some development effort, but the design does not seem to have gone through any usability tests. Still, if they fulfill any of your particular needs, there is of course no harm in using them.