This is a very good question. In order to breathe a creature must have "lungs" and goldfish don't have lungs so "technically" they can not breathe But - just like us - goldfish do need oxygen, which is a gas that we and other animals take out of the air through our lungs.
Goldfish (and other fish) take the oxygen that they need from the water in which they live, and the pink gills, which lie behind their heads and move in and out regularly, are the equivalent of our lungs - drawing oxygen from the water into the fishes bloodstream.
It is important that the water your goldfish lives in is kept clean and fresh - otherwise your fish could get into serious trouble due to lack of oxygen. Changing it's water regularly is helpful - and the water can be kept topped up with oxygen by bubbling air through it using an air pump.
Incidentally, there is a fish (called the Lung Fish) which does have a simple lung and which can breathe out of water for quite some time !
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