APA stands for American Psychological Association. The APA publishes a handbook, called The Publication Manual of the American Psychological Association, that dictates the format and style of papers published in the sciences. It includes formatting rules for the layout of papers and for citations within and at the end of papers. It also has rules for the use of footnotes and endnotes and guidelines for puncutation and grammar.
The APA's guidelines are important because they promote consistency within a discipline. Everyone writing biology papers, for example, uses the same rules. So everyone reading a biology paper knows where to find citations, how to read tables, when to look for a footnote, and so on.
There are other guidelines as well. Two of the most common, aside from APA, are MLA for the humanties and Chicago for journalism. All of these guidelines are updated periodically to make allowances for changes in media. For example, the previous edition of the APA guidelines did not have any rules for citing websites. The latest edition does.