Most WordPress themes use the excerpts in the lists of posts as a small summary of the posts’ content. That allows the visitors to quickly understand which is the main topic of each post before reading them.
When your theme uses excerpts, it is important to set one for each post. The excerpt must be attractive to get the attention of your visitors. That will make them want to continue reading the full content.
In addition, the excerpts are often used as replacement of the full content in RSS feeds. You can get more information about it visiting the WordPress Codex.
You can write the excerpt in the edit page of a post, inside the Document » Excerpt section on the sidebar of the editor.