APA style is commonly used in social science fields such as psychology, economics, education, and political science. Here is a link to the style guide and an online resource for using it. You can also refer to The Longman Hanbook for Writers and Readers which you received as plebes and is available in the Ready Reference section of the Nimitz Library.
The style developed by the University of Chicago and published in the Chicago Manual of Style (CMOS) is used for scholarly publishing in many disciplines, but especially for history and philosophy. Turabian refers to a simplified guide to Chicago style written by Kate L. Turabian for students. Citations are virtually the same, but use CMOS to be safe unless your professor specifically instructs you to use Turabian.
MLA is the preferred style for most humanities disciplines with a couple exceptions (see Chicago style above). The style guide is not online like the others, but MLA does provide an online citation guide. The official source for style information is the MLA Handbook for Writers of Research Papers, and you can also refer to The Longman Handbook for Writers and Readers which you received as plebes. Both handbooks are available in Nimitz Library.
While not employed as often as the other styles in this guide, the styles below may be needed in some situations at USNA. The links will take you the style guide where you can find examples of how to cite different resources.