What 'reserve currency' means is that Central Banks and international businesses hold some money aside for a rainy day. The hold foreign reserves so that when foreign bills become due, if they lack the means to buy foreign currency they can use the reserves. This makes it easier to get loans from foreign banks, because they can see that you have some spare cash you can use in a tight spot. It doesn't have to be the USD, most central banks hold reserves in many currencies as well as gold. This can be sold at short notice and be used to repay loans in foreign currency. The fact that the USD is the main reserve currency means that there is greater demand for USD then Swedish Krona for example, increased demand means the US gets a better price for its currency then Sweden. So the benefit is lower interest rates for the US.
USD are a great reserve currency because there are so many of them, and there is a very liquid market. So a central bank could sell a lot of them in a short time without too many problems.
The gold standard was only one part of Bretton Woods, the other parts are the IMF, WTO and World Bank, which all support the USD by making all transactions dollar denominated. So even though the gold standard has been dropped, Bretton Woods is still functioning as intended, without the hassle of a gold standard.
The only thing that could cause the USD to lose reserve currency status would be a collapse of the US economy, because it is still backed by the IMF, WTO, World Bank and the pentagon. Getting around those things is no easy task, and can only happen very gradually, at the same time most of the worlds GDP supports the USD status.
As to the title question:
Fiat money is currency which derives its value from government regulation or law. It differs from commodity money, which is based on a good, often a precious metal such as gold or silver, which has uses other than as a medium of exchange.
So yes there are many regulations and law that give the USD value, not just in the US but all over the world, you could say the USD is the worlds biggest fiat currency

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