Here are some of the books that I think are must have books for any student's library:
1. Private Pilot Practical Test Standards
This is something you cannot go without on your training to earn your private pilot certificate. This tells you exactly what to expect on your Private Pilot checkride and what books and Advisory Circulars the Federal Aviation Administration expects you to know.
This can be purchased at any pilot shop or can be found online at:
2. Airplane Flying Manual
This is highly recommended for students because it describes flight maneuvers, how to inspect aircraft, and much more necessary information for flying.
This can be purchased at a pilot shop or found here:
3. Pilot's Handbook of Aeronautical Knowledge
This one has been recently updated as of 2008 to include more information on glass cockpit instrumentation.
This can be purchased or found online here:
4. Federal Aviation Regulations/ Aeronautical Information Manual
These are the rules and regulations by which to fly by. The Federal Aviation Regulations tell you exactly how to earn your certificate, the maneuvers required to be proficient, and how many hours you need (Part 61), and also how you can lose it if you aren't following the rules (Part 91).
There are two major publications, one through ASA and the other through Jeppesen. The choice is up to you one which one to get as they both have the regulations you will need.
***Something to note about the FAR/AIM*** Just because you have the book does NOT mean you have the most recent regulations. The publishers put out updates to the regulations that you can receive email or mail updates.
These are the majors I have time to write about right now, there are more to come!