Live rock is just that, rock with a variety of living creatures living in, on, around and of course off of it. The true definition of reef is a submerged ridge of rock or coral near the surface of the water. however when you wrote reef I believe you are talking about when people say "reef tank" meaning a tank with corals and invertebrates. In the saltwater end of the hobby people usually keep:
FO - Fish only,
FOWLR - Fish only with live rock, or Reef - which is usually live rock, reef safe fish and invertebrates, and corals. You will find live rock in any reef environment, naturally, and in most captive situations as well.
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