What is the purpose of this test?
The test (known as a review audit) is designed to help new reviewers hone their moderation skills, while nudging more experienced users that don't seem to be paying close attention to what they're reviewing. Some people were in the habit of up-voting everything they reviewed without regard to quality; extremely low quality posts with a high score are very problematic in a system where the best information naturally rises to the top, so "audit" tasks are mixed in to make sure folks are paying attention.
This is currently active only on Stack Overflow, Server Fault, Super User, Programmers, Ask Ubuntu, Mathematics, and Puzzling.
How does the audit work?
First Posts, Late Answers and Low Quality - The test presents a reviewer with a post that was previously removed by a moderator and known to be of extremely poor quality (flagged as spam, not an answer or very low quality) or high quality (many up-votes, no down-votes, etc).
Close Votes and Reopen Votes - The test presents a reviewer with a question which the system has determined should be open or closed based on various criteria (highly-voted with no close votes, quickly closed with no answers, etc).
Suggested Edits - The test presents a reviewer with a randomly generated edit utilizing the Markov chain algorithm to randomly insert various words that may appear to be useful at first glance but actually make no sense and are meant to vandalize the post.
If a user fails a review audit, they are presented with a message that they didn't pass and encouraged to pay more attention in the future. If a user passes the test, nothing happens and they continue reviewing. In some review tasks, they are informed that they passed and thanked for their time and feedback.
The result is then recorded for analysis. Only diamond (♦) moderators or Stack Exchange employees can see the data; it is not accessible publicly even to high reputation users.
What happens if I fail?
For most, nothing. You'll be given some guidance on why the post was inappropriate and what you could have done instead, and then allowed to continue reviewing.
Repeated failures, especially in short succession can lead to your review privileges being suspended for a week or longer, depending on the frequency of the failures. Stack Exchange is continuing to analyze the data to determine the best action to take in these cases.
Do review audits count toward the Reviewer badge?
Passed audits count as a successful review toward the Reviewer badge. A failed audit does not count toward the badge, but does not decrement your progress; it simply adds none.
Do robo-reviewers dream of electric sheep?
Robo-reviewers don't sleep. They wait.
The information contained in this answer is subject to change as the review system is further refined. Changes that impact the mechanics of the test, or consequences of failing it repeatedly will be reflected in this answer as they're made, but not necessarily immediately after they're made.