Economic and Game Theory
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The idea behind mechanism design is very simple. A mechanism designer is the one who designs the "rules of the game". For example, imagine that a famous drawing by Leonardo da Vinci has been in your family possessions for generations. Now you want to sell it, but you don't know what its value is, that is, you don't know for how much to sell it. The only thing you know is that if you announce to the world that you possess this famous drawing and are willing to sell it, then only the people who really value it will call you to let you know they are willing to buy it. (Technically, you also know the distribution of the buyers' private valuations, but not their actual valuations). What should you do? In other words, your question is: "How am I going to sell it? What is the best way to sell it?"
The answer is provided by the mechanism design theory. You call a mechanism designer and tell her your problem and ask her for her help. She'll tell you the following:
(1) OK, I'll help you. I need to find out who is the buyer that values the drawing the most. Whoever he or she is, the drawing should be sold to him or her.
(2) But how can I know who this buyer is?
(3) My suggestion is: announce to the world that you have this drawing and that it will be auctioned. Announce that it will be a second-price sealed-bid auction. What is it? Well, a typical buyer come to the auction house and submit his valuation (his bid) in a sealed envelope. You collect the sealed bids of everybody. Then you open them. The winner is the person with the highest bid. However, the winner will only pay the second highest bid.
In what sense this is a mechanism design problem? The mechanism designer told you how to sell your drawing. In particular, she recommended a second-price sealed bid auction. She told you the "rules of the game", the "mechanism". Moreover, the mechanism design theory shows that this mechanism induces the buyers to bid their true valuations. This is called "incentive-compatibility condition". This is amazing!
What are the general features of mechanism design? People have private types, that is, some private characteristic that only that person knows, and nobody else does. In our example, the type of a buyer was his/her valuation. You want to implement a particular outcome that depends on these private characteristics. But you don't know them. Then you announce the rules of the game: This means: You ask people to send some "message" to you. You collect all individual messages, and implement the outcome according to the prespecified rules (the mechanism) you announced. The question is: What is the best "rule" (mechanism)? In the book you have read you saw something called "revelation principle". Roughly speaking, this means: It's OK if you restrict yourself to a incentive-comaptible mechanism that asks people to tell you their private types (not a general message). Then the mechanism design theory will tell you that you have to use a incentive-compatible mechanism.
Suppose you think of an economic problem in the real world. Maybe you want to auction a painting, but don't know its value for the market; maybe you work for the government and want to build a bridge, but you don't know how much people nearby value it and how to tax them. You can always formalize these problems as mechanism design problems. Then you just have to find out what is the incentive-compatible mechanism (a mechanism that induces people to tell the truth, their true type). Sometimes you may also requires that the mechanism be "individually rational", which is a fancy expression for "a mechanism that induces people to participate in the game".
I hope it helps you.