Generally, the author of a work owns the copyright. Copyright is granted automatically when a work is created; authors do not need to register to be protected by copyright. However, registration is required in order to bring a lawsuit for copyright infringement.
There are some exceptions to this, such as the work for hire doctrine, which relates to works made during the course of an employment contract. Works made for hire include works made by an employee as part of their job, or works commissioned as part of a collective work, provided that the author has signed a contract to this effect.
For works made as part of a collective work such as an anthology, absent an agreement otherwise, authors are presumed to own their contribution. Each author is able to exercise their exclusive rights over their work.
For works made by two or more authors as part of a joint work, such as an article with multiple co-authors, authors are presumed to be co-owners of the copyright. Joint authors retain non-exclusive rights over the work. Joint ownership can be complex, and authors may want to discuss copyright ownership at the outset of a project.
One way to determine ownership of a work is to look for a copyright notice. Copyright notices are not required to establish copyright, but they are present on many published works. You can also check the Copyright Catalog, or ask the United States Copyright Office to conduct a search for you [pdf].
Copyright is often referred to as a "bundle of rights," meaning that it is a collection comprised of different rights. An author can transfer none, some, or all of the rights granted by copyright. When an author publishes a work in a book or journal, they may be asked to transfer all of these to the publisher. However, there are other options which allow them to retain some of their rights, such as the right to make the article publicly available. There are many resources offering practical guidance for authors interested in retaining rights.
Authors may also choose to use a Creative Commons license to share their work. Creative Commons licenses are public copyright licenses that are more permissive than traditional copyright agreements. Creative Commons licenses allow authors to retain copyright while creating a standardized way to grant others permission to copy, distribute, and use their works, rather than granting permissions on a case-by-case basis.
For works created in the U.S. after 1977, the copyright term usually lasts for the life of the author plus 70 years. After that, the work generally passes into the public domain. There are many different factors, including the date of publication and registration; this chart from Peter Hirtle is an excellent resource for determining copyright term.
Public domain refers to works that are not covered by intellectual property laws, and are owned by the public. Public domain works can be used by everyone, but are not owned by anyone. The University of California, Berkeley School of Law has an extensive guide [pdf] on determining if a work is in the public domain. Authors can also dedicate their works to the public domain using a CC0 license.