Who grades the GRE essays?
For the GRE AWA, each essay receives a score from at least one trained reader. What do we mean by "at least?" Well, by default each essay is looked at by the human reader and an e-rater. If they agree very closely, then the average of those scores is your score. If the human and e-rater do not agree, however, another human score is used instead of the e-rater, and your score becomes the average of the two human scores instead.
What are they looking for?
ETS says that essays are graded "using a six-point holistic scale." In holistic scoring, the professional readers are trained to assign scores on the basis of the overall quality of an essay (both its quality as writing and its quality in response to the assigned task, so the instructions matter). ETS also tells us that "the primary emphasis in scoring the Analytical Writing section is on your critical thinking and analytical writing skills rather than on grammar and mechanics."
What score level(s) is(are) possible?
Each essay gets a score. The final scores on the two essays are then averaged and rounded down to the nearest half-point interval on the 0–6 score scale. You will only have one single AWA score reported. You can see the individual score percentile rankings on this ETS PDF.
How do I score my own essay?
We have distilled the official ETS documents into an easy-to-read rubric. Head over to this article for the rubric to grade your essay and some tips: Score your GRE Essay.