A research (or term paper) is an extensive research which is a culmination of a significant study or a body or work a student necessarily faces in his/her academic career. Usually carried out at post-secondary level, this kind of writing requires a more abstract, critical, and thoughtful level of inquiry than one can be used to. By writing a research/term paper you actually show your mastery over the subject, your ability to incorporate theory into practice, to analyze, compare, think critically and go beyond the obvious.
APA is short for American Psychological Association; an organization, which has elaborated a set of referencing rules for different fields of study, e.g. education, business, various social studies and, naturally, psychology. Hence an APA term paper means that this particular piece of writing is to be formatted and referenced in full accordance to the specifications, set by American Psychological Association.