The key to playing chess is understanding checkmate. One of the best instructional books of all time, Bobby Fischer Teaches Chess uses chess diagrams to offer instructive problems that become progressively more difficult. Do these exercises and you will improve your game.
In The Chess Tactics Workbook, Al Woolum presents 160 pages of puzzles (6 per page) concentrating on mates in 1, 2 or 3 moves. Lots of great practice for the beginning student.
Because these books are not filled with illustrations, they are great choices for beginners of any age.
Grandmaster Murray Chandler is most known for How to Beat Your Dad at Chess (covered below) but his earlier book Chess For Children is a great resource for early elementary readers with large type and illustrations. The fun stories might need an explanation... beginning 5, 6 and 7 year-olds will need to be told that the alligator is pretending he beat really famous chess players. That said, the instruction is clear and easy to follow and starts with the very basics of how to set up the board, how to move the pieces, and rules before getting into some basic tactics and strategy.
This volume can then be followed up by Chess Puzzles For Kids which works well with older elementary kids as well. Chandler offers 100 different puzzles; the reading level is a little further along than Chess for Children.
Teaching the basics is easy with Jeff Coakley's Winning Chess Strategy for Kids. Covering the rules, basic mates, and elementary tactics. It then leads the student through a whole range of advanced strategies, including piece development, pawn structure, and attacking the castled king. Recommended for students 7 to 13.
For extra practice Winning Chess Puzzles For Kids provides exercise sheets with more than 1000 positions, covering mates in one or two moves, and simple tactics such as forks, pins, and discovered checks.
For Advanced Beginners
A chess book for everyone, from eight to eighty, How to Beat Your Dad at Chess explains numerous ways to beat a stronger opponent (be it a friend, clubmate - or Dad!) GM Murray Chandler includes 50 checkmates. - Recommended for older elementary and junior high ages.
How to Beat Your Dad at Chess works well with Chess Puzzles For Kids, covered above, and with Chess Tactics for Kids. This volume features 50 tactics focusing on recognizing the patterns that occur most frequently in real-life play.
For Advanced Beginners/Intermediate
Continuing with the Jeff Coakley series, Winning Chess Exercises for Kids clearly explains tactics, strategy, and terminology in simple language graced by lots of illustrations and exercises.
For extra practice Winning Chess Puzzles For Kids Volume 2 provides more exercises at a slightly higher level than Volume 1.
For Teachers and Older "Advanced Beginner" Students
Of course our first recommendation is Great Moves: Learning Chess Through History, by Sunil Weeramantry, Alan Abrams and Robert McLellan. Written for students with a middle school reading level and above, this text blends chess instruction with history providing portraits of many of the players and a contextual overview of the world when chess was developing to the game we know today. Chess instruction begins with a review (or an introduction, for new players) of the basics but presented in the context of when these rules and tactics were first published or widely adopted, and advances to analyze games from the leading players of the late 1800s whose play set the standard for today's grandmasters.
In Logical Chess: Move by Move, Irving Chernev shows 33 games in detail, explaining the moves of the masters and showing readers where to place the chess pieces for the best results. A perennial best-seller, this book is recommended for older children (and for adults) as well as for teachers.