Morpha itself, refers not to the transparent muscle-like tentacle as its Ocarina of Time official artwork suggests, but the vulnerable red nucleus, often safeguarded under the muck of the thick liquid. This nucleus can manipulate the water in any way it likes, creating one (and eventually two) large water tentacles as the battle goes on.
To defeat it, Link has to pull out its nucleus with the Longshot and attack it with the Master Sword. After enough strikes Link must dodge even more tentacles that could catch him off-guard. If Link still manages to pull through and slash at it a few more times, Morpha will liquify.