Archive of Our Own (or AO3) is a popular non-profit open source database for fan fiction and other fanworks that allows people to share their creativity online without the fear of copyright infringement. Created by the Organization for Transformative Works in 2008, AO3 is known for its extensive tagging system and overall organization. Users contribute to the site by creating an account, writing fanfiction, fan art or podfics, and submitting them via an uploader. The fics are then searchable and browseable, and writers receive kudos when their works are read and enjoyed by others on the site.
In a more general sense, an archive is something that holds historical records and materials for use in the future. For example, a library might have an archive of books and papers that are kept in a special room because they are important for researching the history of the library itself. The term is also used in some businesses to mean the storage area for documents, such as invoices and letters, that are not yet processed or sent out for payment.
For most of us, the most common usage is similar to a filing cabinet. We tend to think of an archive as a way to organize files in a specific and organized manner for use later, which is more inline with what archivists would say. The most important element of an archive is that it is still accessible and usable, which can be difficult in some cases.