Page excerpts are slightly different than post excerpts. In my three years of development with WordPress; I have only seen page excerpts a few times.
Page excerpts may be used in custom queries; to show relevant matches to a search (for example).
There are many plugins available in the WordPress Plugin Repository which manipulate page excerpts and add extended functionality.
Note; not all themes will use a page excerpt area. It is up to the theme author when designing the theme to determine if a page excerpt editor is needed.
Page Excerpt Editor
If your theme does indeed use page excerpts; then there will be a box under the main content editor when creating/editing pages.
NOTE: If you do not see the page excerpt area; you may have to enable it from the Screen Options tab. Scroll to the top of the post/page you are creating/editing; and look for the “Screen Options” tab at the top right of the page.
Click the “Screen Options” tab; and a window will slide down. Check the “Excerpt” checkbox, and close the tab.
WP Edit Pro Page Excerpt
Notice in the screenshot above how the editor looks rather… bland. Where are all the cool buttons and ability to insert media?
The ability to add rich text to the excerpt area is not available by default. With WP Edit Pro, we can enable the standard content editor inside the excerpt area.
Go to the WP Edit Pro admin settings page -> General tab.
Check the option; and then revisit the pagecerpt editor. It is now a duplicate of the main content editor.