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A coral reef is a calcium carbonate (limestone) formation produced from the build-up of coral skeletons. Coral is a colonial organism, but they are also individual. Corals are individual organisms, and they group together and form a colony and create reefs. How does this happen? Corals all have calcium carbonate skeletons, which they secrete themselves. They group together, and their old, dead calcium carbonate skeletons form coral reefs. The reefs grow by the coral spreading and continually growing calcium carbonate. Reefs are commonly classified and grouped into three main groups: Atolls, barrier reefs, and fringing reefs. 
Coral Reefs are not grouped within the normal Taxonomic hierarchy like most other organisms. Why is this? Because they aren't organisms, they are biomes. A biome is a community of organisms, commonly referred to sometimes as an ecosystem. The coral all make up the colonial organism and create the calcium carbonate, and create the reef. Where they create the reef defines how they are... defined. There are three main classifications of reefs, and all of the leading sources disagree on how many classifications there are, but in every source, these are dominant:
An atoll is a reef, circular, usually surrounding an island. They usually have a lagoon in the center, and the paths to the open ocean are usually through channels.
These reefs are the most common. Fringing reefs are always connected to the mainland, hence the name, fringing. They are younger reefs, and give way to both of the other two reefs. 
Barrier reefs are reefs that are close to the mainland, or far from the mainland, but still have the mainland in sight. They are not connect to the mainland in any way. The largest barrier reef in the world, as well as the largest reef in the world, is the Great Barrier Reef in Australia.
Reefs themselves do not reproduce; they are built by the colonial coral. However, the coral themselves do reproduce, and they reproduce asexually, mainly by fragmentation (by breaking apart and becoming two separate corals), and by mass spawning. The polyps release large amounts of sperm and eggs and they float to the surface to fertilize The coral may reproduce by budding, the process where the organism asexually reproduces itself, and attaches the new organism to itself before becoming free again. Reefs are a biome, so they themselves do not reproduce, but what they are made up of does, constantly. 
All coral reefs are found exclusively in shallow waters teeming with life, with a few exceptions. Most reefs are found in tropical waters, and are photosynthetic. They host a variety of organisms, such as anemones and clownfish, and reef sharks.
Coral reefs are undergoing serious conservation attempts right now. They are endangered for a number of reasons. Some natural reasons are: tidal immersions, typhoons, monsoons, climactic changes, hurricanes, and any other radical changes in the habitat. Among the human reasons are: pollutants, ocean acidification, dangerous fishing techniques (cyanide and dynamite fishing), overfishing, dust outbreaks and dust storms, and unregulated tourism. Pollutants and dangerous fishing techniques are linked because when you use cyanide, the reefs die, and they are forever harmed. Reefs are very stable organisms, and if they are strong at the time of the harm, they will most likely survive and come back, but with all of these factors, that is impossible. Ocean acidification is a problem because of the riding CO2 gases in the atmosphere, causing more CO2 to appear in the oceans, causing ocean acidification. Overfishing is a problem because if you take away the fish that feed off the other fish in the reef, there is an unbalance of power and in the food chain. Another pollution problem is agriculture and sewage runoff. Dust outbreaks and dust storms are a problem because they make the reefs unhealthy, making it hard for them to survive. Unregulated tourism is a smaller problem, but still a problem nonetheless, because they don't know how to treat the reefs, and harm them during common tourist activities such as scuba diving. It is estimated that we have lost over 10% of the world's coral reefs already. 
Great Barrier Reef
The most popular reef by far is the Great Barrier Reef found in Australia. The Great Barrier Reef is actually made up of 3,000 individual reefs and 900 islands! Not only is this reef the most popular, but it is the largest reef in the entire world. It is around 1,600 miles long, greater than the distance between Seattle, WA and Los Angeles, CA.  The Great Barrier Reef is the largest single structure built by living organisms, and can be seen from outer space! 
Belize Barrier Reef
The second largest reef in the world is found in Belize, South America. It measures at around 185 miles, and is Belize's top tourist destination. It is the longest barrier reef in the Northern and Western Hemisphere, and surrounds a caye called Ambergris Caye. It surrounds the island, protecting it from crashing waves and brings and abundance of life.
What's on my shoe?
Scientists used to think that coral reefs were formed millions of years ago, contrary to popular belief nowadays. Why has present belief changed? Scientists recently discovered a piece of coral growing on a man's shoe he had left in water. Shortly thereafter, scientists discovered something else- growth rings, very much similar to a trees. Using the growth rings and the evidence of the corally shoe, scientists are starting to realize just how fast coral can grow, and how fast their debate over their millions of years is crumbling. They analyzed samples from the Great Barrier Reef and all of the reefs that make up the Great Barrier Reef using these new methods. They estimate now that the Great Barrier Reef could have been formed about 3,700 years ago, not eons. 
Evidence for an old earth?
Many have claimed that it requires long ages for coral reefs to form. They claim that this falsifies the young earth view of origins, but is this true? Does it stand up the critical review? To put it simply, no. The following are reasons why.
A few have pointed out that many so called coral reefs are nothing more than carbonate platforms. These formations were most likely formed during the Flood. The ACTUAL coral is only a thin layer on the top. (Oard, M.J. The paradox of Pacific guyots and a possible solution for the thick ‘reefal’ limestone on Eniwetok Island, CEN Technical Journal 13(1):1–2, 1999. Cited in Geology and the Young Earth by Tas Walker)
It seems the fossil coral reefs of the great lakes follow this trend. This reef heavily contradicts the quality of many other known reefs. 
In 2001, researches found that Australia’s Great Barrier Reef isn’t 20 millions years old, as many old earthers believe, but 600,000 years old, if you assume old earth dating methods. They then theorize that there was some global reef initiation event. (Science, 25 May 2001, pp. 1451-1453.)
But even this date is too old. According to Paula Weston:
|“||From strongly correlated historical records of river run-off and flooding, the researchers determined that the coral colony in question had taken only 118 years to grow 1.8 m (6 ft). This gave them a new method for determining the age of all reefs that make up Australia’s Great Barrier Reef. Based on this methodology, the outer barrier reefs (those furthest from the Australian coast and in the deepest water), which are about 55 m (180 ft) thick, would be less than 3,700 years old—not millions of years old as has been believed.||”|
The rate used by coral reefs dating are only from measurements of surface coral, but don’t take into account that ultraviolet light all inhibit growth is coral and that reef growing organisms can die easily from conditions found on the surface, i.e. air during very low tides.
We have measured coral growing several times faster then excepted. Some have reported a maximum rate of growth of 414 mm/year. This would allow for the development of a 1405 m reef in less than 3400 years.
Reef growing organisms may not be the only factors in the growth of coral reefs. Reefs can act as a filter, trapping some carbonate from the seawater passing through. 
- Oard, M.J. The paradox of Pacific guyots and a possible solution for the thick ‘reefal’ limestone on Eniwetok Island, CEN Technical Journal 13(1):1–2, 1999. Cited in Geology and the Young Earth by Tas Walker
- FOSSIL BINDING IN MODERN AND ANCIENT REEFS by Lance T. Hodges
- CORAL REEF GROWTH by Ariel A. Roth
- Coral: animal, vegetable and mineral by Paula Weston. Creation 25(1):28–32December 2002
- Great Barrier Reef Surprisingly Young Science, 25 May 2001, pp. 1451-1453
- Wikipedia Coral Reef
- Enchanted Learning Coral Reef
- National Center for Ecological Analysis and Synthesis, University of California, Santa Barbara Coral Reefs
- NOAA Coral Reefs
- NOAA What Are Coral Reefs?
- NOAA Hazards to Coral Reefs
- PBS Anatomy of a Coral Reef
- Starfish Reef classification
- University of Puerto Rico Reef Classification
- Coral Facts Coral Reproduction
- Wikipedia Great Barrier Reef