In the Caribbean, there are 15 or so kinds of moray eel. Based on my antidotal observation, the green moray is the most common. Green moral eels tend to grow larger than their cousins so perhaps their ability to grow larger gives them an advantage in the wild. Based on my personal observation around the Caribbean, the spotted moray is the second most common moray eel to see.
Here is an example of a spotted moray eel. 
Ordinarily, moray eels are relatively unaffected by the presence of divers. Quite often divers can get relatively close to moray eels before the eel will withdraw into the reef to avoid contact with a diver, but not always. 
Sometimes during the daytime, eels will only peer out of their hiding holes in the reef and you have to be fairly alert to spot them. 
Sometimes if you are very fortunate you will see a moray eel freely swimming about the reef. Generally speaking, you are more likely to see a free swimming moray eel during a night dive when they are hunting for food than you are during the day.
The relationship between divers and eels does change if there is food that is present or the eel believes that there may be food available. When food is present, the moray can become much more interested in the food than anything else. We were diving in Belize on the Turneffe reef when I saw a moray become much more interested in food than anything else. Our dive leader had speared a lion fish and were swimming along a wall when a moray caught the sent of the dead lion fish which was still on the spear. Here is how that interaction played out. Nevertheless some moray eel have no interest in lion fish as food even when dead.
Sometimes even if the feeding by has not occurred by humans for quite a long time, a moray eel will continue to look for a free meal. occasionally, the interaction between divers and an eel can be almost comical. Unfortunately, feeding wild animals can change their behavior. That applies not just to moray eels but to other animals as well.
Nevertheless, getting to see a free swimming moray eel, particularly in the day time, is quite a treat. And, while they are technically fish, moray eels are quite different from most fish that you see in the ocean.
Here is a short clip of some moray eels I have seen recently.
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