Breaking is intended to power and precision - the ability to place sufficient power precisely enough to break a solid object. We break two things - boards (#3 common pine, 1 x 12, cut in 12 x 12 pieces) and tiles (concrete roofing tiles, roughly 16 x 16). We do not use spacers - as others have stated, spacers make breaking much easier, so we don't use them.
In TKD (at least the ITF TKD I started in, and the YCTA, my current association) breaking begins at yellow belt; however, children are not allowed to complete hand breaks at all, or foot breaks that might damage growing bones, nor are hand breaks allowed for anyone whose job might be jeapordized by damage to their hands - a friend of mine, for example, is a dentist, and is not allowed to do hand breaks. The number of boards is determined by the student's age, rank, and size.
Breaking occurs in 3 approximate categories: power, technique, and hand. Power breaks are just what they sound like: breaks using techniques such as side kick or back kick, with the intention of breaking as much as possible. When my sahbum tested for his VI Dan, he broke 8 boards with a standing side kick from 1" off the front board - pure power, like Bruce Lee's 1" punch. Technique breaks are generally (but not always) foot breaks, using techniques that are technically much more difficult - like twisting kick - to break smaller numbers of boards, to demonstrate technical proficiency. When I tested for IV Dan, I broke with a head-high twisting kick, less than a foot from my own head - but only 1 board. Foot breaks are generally to boards. Hand breaks are just what they sound like - breaks done with the hand. They are done with either boards (basic) or tiles (advanced).
There are pros and cons - as well as strengths as weaknesses - to both boards and tiles. Boards have a lower tensile strength than tiles, and have a grain, both of which facilitate breaking; tiles, on the other hand, while stronger, are also more brittle. Boards must be struck fairly precisely, and along the grain of the wood; tiles are homogeneous in their structure, and must be struck with sufficient power, but are more brittle than wood.
Breaking is one piece of demonstrating technique and power - but only one piece. It can be a lot of fun, and can build confidence in students and impress observers - but while it does demonstrate competence, it is no more indicative of overall ability than other other isolated facet of a martial art.
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