Categories are used to segregate posts into related material. If you create a category then assign that category to multiple posts, then WordPress can provide you easy access to all those posts in that category. That is the archive ability of WordPress.
Don't be confused by what an archive is--it is just a generated display of your posts at that moment. You don't do anything to create the archive, that's an automatic thing WordPress does for you. Usually, archives are date, category, tag, or author, based.
Access to archives is typically presented via links in a sidebar under an Archive (date based), Category, or Tag Cloud, heading. Widgets, or Template Tags, such as wp_get_archives(), wp_list_categories(), wp_tag_cloud(), and wp_list_authors(), are the constructs used to present links to users to visit your various archives. The process of placing code in your Theme's Templates is explained in Stepping Into Templates and Stepping Into Template Tags.
Once a user clicks on a Category link in the sidebar, the display of those posts can be controlled by a Category Template. Other Templates, such as Author Templates, and Tag Templates, are available if you set them up. These Templates can be coded via Template Tags such as the_title(), the_content(), or the_excerpt(), to display just a post title, the full content of the post, or just an excerpt of the post.
Also, it is important to understand the Template Hierarchy, as that is how WordPress determines what Template to use to render the posts for reading by your readers.
If a user visits a Category archive, then clicks on a given post title in that Category archive, the display of that single post is again presented by another Template, and again, the Template Hierarchy determines what Template displays that single post. Finally, that single post Template can be coded to display just the title, the full post content, or an excerpt.