Rule: In general, use the author's or speaker's full name the first time you need to mention her or him. After that, use her or his last name.

First, let's look at an example. Imagine you're reading an essay written by Zadie Smith about public libraries. You'll need to refer to her throughout your writing. (Note: The instructions state that Smith is a female; you should be sure to refer to her appropriately when you write.)

I have read essays with several different methods of referring to Smith. The first, and best, is simply to say "Smith." For example, "In her essay, Smith argues that public libraries are important centers of culture." This is the correct way--using Smith's last name (also referred to as the surname or family name).

Examples
Best: "Smith claims that libraries exist for altruistic reasons."
This is correct--here we use Smith's last name.

Avoid: "Zadie claims that libraries exist for altruistic reasons."
This is not standard. To my ears, when I hear this (or read it, as it were), I feel like the author is a friend of the writer.

This mistake isn't terribly common, but I do see it, so if you're using this format, you should know it's not considered standard for formal essays.

Avoid: "Zadie Smith claims that libraries exist for altruistic reasons."

I also see this form from time to time--using the writer's full name. We generally don't do this in formal writing mostly because it's just too time-consuming to write out the full name.

I'm working on a series of articles related to really common mistakes, and this is one of the first in the series.