One of the four elements of Tae Kwon Do (besides sparring, style forms and self-defense) is the break-test. It is a obligatory part of the black-belt exam and is s popular element of Tae Kwon Do demonstrations.

Why?
Breaking an object is a good way to practice concentration, power, focus, speed and precision on non-living objects, without injuring oneself or another. It is very important to realize that a proper technique is needed and a breaking technique within your limits, because without it you can easily injure yourself, sometimes even permanently!
Practicing breaking objects helps you to realize that your body itself is a very strong weapon. It also helps you to understand that during practice with a partner, you have to be very careful.

How?
The material that is most often used for breaking techniques is wood since it can be easily broken with either hand or foot. Other materials include bricks, tiles and sometimes even baseball bats!
Breaking objects can be performed with any rigid part of the body. In Tae Kwon Do, the most common are the hand or the foot, but breaking can also be done by using the elbow, the knee and even the head.
In order to break an object, it is best to start light. Use an object that is easy to break, instead of directly trying to break a thick board, and start with a technique in which you feel confident. Use your techniques as they are taught to you and try to be as relaxed as possible. When starting practicing breaking techniques, it might help you to focus a few inches behind the actual point of impact.

Physics
Dry boards are more brittle than wet ones and will break more easily. If a board does not break, a large force is transmitted back to your body for a relative long time. This might hurt :o)
Break boards with the grain. It is much easier!
When breaking a board, make sure that the persons who hold the board have a firm grip. If the board moves during your technique, it will soften your technique.
Other factors not covered:
Angle of strike
Size of attacking tool
Impulse = force (delivered by the strength of t=your muscles) times time (delivered by the speed of your movement). Therefore, the shorter your contact with the board, the bigger the force component will be and therefore, the easier the board will break.
Momentum = Mass (weight) x velocity (speed in a certain direction), The change in momentum is the momentum when you hit the target minus the momentum when you come to a stop, which is determined by weight and speed (== impulse??) The force applied to stop a movement determines how quickly it is stopped, since we can say the  mass used to create the momentum will not change.
In order to break a board (or any kind of material), you must cause a shearing moment in the board that is larger than the critical moment for that type of material. That shearing moment is when you try to break a board, the board itself is supported as both sides. If you perform your breaking technique well, you will hit the board in the center which leads to an equal distribution of force on the two parts. Both parts will supply a reverse force of half the initial force.
When the force meets the board, the top of the board will be in a state of compression and the bottom will be in tension. This will produce a torque on an axis through the middle of the board. If the torque is great enough the board will break
Besides force other elements that are important are power and pressure.
The thicker the board, the harder it is to break the board. That is why often multiple smaller boards instead of one thicker one are used.