The term "author's rights" refers to the intellectual property rights possessed by the author or creator of a work, which are generally governed under copyright law.
As the author/creator of an original work, you own the copyright to that work as soon as it is fixed in a tangible form (such as print or electronic text or images), which gives you the exclusive right to copy and distribute your work. Specifically, under US copyright law, the owner of the copyright has the following exclusive rights:
See: Kernochan Center, Intro to Knowing Your Rights.
In order to publish a book, journal article, or other work, a publisher must obtain the right to distribute your work, which is generally granted via a publication agreement.
What rights are transferred to the publisher in these agreements can vary widely, including:
See: Georgetown University Library, Negotiating Your Contract.