When editors or teachers ask you to write in APA Style®, they are referring to the editorial style that many of the social and behavioral sciences have adopted to present written material in the field.
A physical copy of the Official APA Publication Manual can be found here at Kirk Library and is available for your reference.
About APA Style®
The best scientific writing is spare and straightforward. It spotlights the ideas being presented, not the manner of presentation. Manuscript structure, word choice, punctuation, graphics, and references are all chosen to move the idea forward with a minimum of distraction and a maximum of precision.
To achieve this clarity of communication, publishers have developed rules of style. These rules are designed to ensure clear and consistent presentation of written material. Editorial style concerns uniform use of such elements as
Source: APA Style Official About Page
There is a great tutorial that you can take as an introduction to APA style if you're unfamiliar with this format or need to refresh your skills: