The “saturated” part refers to how “saturated” the carbon chains are with hydrogen.
At a simple level, fats have some long chains of carbon. If the carbon molecules are all joined together with single bonds only, then the carbon atoms will be “saturated” with hydrogen – in other words, they are bound to as many hydrogen atoms as possible. This makes the fats hard at room temperature.
If there is one (and only one) between c/two carbon atoms in the chain, then that chain of carbon will hold less hydrogen, and so is “unsaturated”.
If there are multiple double bonds between various carbon atoms I. The chain, then it holds even less hydrogen, and is “polyunsaturated”.
If memory serves, the more double bk ds, then the less viscous the oil, however someone who has been involved in chemistry more recently than me can clarify better.
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