Intrinsic value is value a thing has regardless of the holder's ability to swap it for something else. So a US hundred dollar bill has the same intrinsic value as a monopoly money hundred dollar bill, in that both are a piece of paper that could be used to wipe one's arse, or start a fire, or write a short note upon (though a similarly sized blank piece of paper would be more vsluable for the latter purpose).
Extrinsic value is the additional value a thing had due to the holder's ability to swap it for something else. Extrinsic value arises because society contains feedback loops, in which people can be persuaded to swap the thing in question for more and more other things, on the basis that the increase in value will continue - the thing is valuable because other people value it, and other people value it because they see it as valuable. The initial seed for such feedback loops can be the acceptance of the thing in payment of tax (as with US dollars, for example); Or its appearance and prettiness (as with diamonds or tulips).
The error of believing extrinsic value to be intrinsic is just the same error made by all religionists. They like something, so they declare it sacred, and denounce anyone who points out that it's just a book, or a statue, or a flower, or a lump of soft yellow metal.
Gold has enormous extrinsic value, and likely will for the indefinite future, because adherents of its religion are legion. But anyone who thinks it has intrinsic value should ask themselves whether they would wish they had some gold, if stranded on a desert island.
The acid test for intrinsic value is whether a solitary person with no exchange interactions with others would want the thing in question. It's situational - if you have a beach-buggy or a generator, gasoline might be more intrinsically valuable, while if you don't, it is much less so. Ultimately, intrinsic value pits a floor under the collapse of a speculative bubble. If people stop valuing dollar bills as dollars, their value as paper stops them from becoming truly worthless.
Ironically, gold might be more intrinsically valuable if it wasn't used as a store of massive extrinsic value (via the greater fool theorem); It could be more widely used in electronics, for example, if only it were not so expensive. It's intrinsic value is therefore hard to determine, but is unquestionably a tiny fraction of its market value. It's more valuable than paper, just because metal that doesn't readily corrode has several good uses.
One of the things that makes anything a useful form of money is a low intrinsic value, coupled with a high extrinsic value. As such, gold makes a fairly okay choice to use for money, but fiat money is better.
Of course, if we redefine the word "intrinsic" to mean "extrinsic properties that have exhibited extremely long life and high durability", then everything
@Swammerdami says about gold is true - but hugely confusing to anyone who uses "intrinsic" to mean common what it commonly means in English.
It certainly confiued the crap out of me; I genuinely thought he believed gold to have most of its current market value regardless of access to a market or exchange. Which would be utterly absurd. Unless he can eat gold.