Copyright registration and the rights conferred under copyright are sometimes conflated. Copyright law in the United States covers certain rights automatically (such as rights of reproduction, performance, distribution, and the creation of derivative works) and isn’t dependent on registration. Copyright registration in the US—the process of applying for copyright with the Library of Congress copyright office—gives stronger protection to copyright, particularly in regard to litigation-related damages. Many American university presses specify that the press will register the copyright for your book after publication, but you should check your contract to see if that obligation is specified.
US publishers often specify in their agreements that the copyright will be made in the name of the publisher (or, in the case of some university presses, in the name of the university). For certain types of scholarly publications, such as edited collections, registering the copyright in the name of the publisher is simpler to process in the registration system. In addition, copyright in the publisher’s name makes it easer for the publisher to defend the copyright when it is infringed and for third parties to know whom to contact initially for permission for possible reuse.
The name the registration is placed under does not confer certain rights to either the author or the publisher—those details, including the right to publish, are spelled out in your contract, so you cannot presume that, for example, authors whose works are registered under their own names retain certain subsidiary or publishing rights. Regardless of the name on the copyright registration, copyright protection under current US law lasts the lifetime of the author plus seventy years (with different terms for anonymous and pseudonymous works).