The gold standard in the US was actually kind of complicated.
You had the government issuing notes off of gold, and private banks issuing bank notes off of gold...but also banks issuing notes/deposits off governments notes as well. A pyramid on a pyramid.
Neither was 100% backed and both were vulnerable to runs.
In 1933, FDR no longer let the public redeem gold for dollars. A default of sorts...but we were never on a true gold standard! We always had more promises than gold. The same with the banks. Given that gold used to be equivalent to dollars (as the Fed used to exchange the two), banks would use dollars as reserves for convenience...but soon they became the monetary base.
Now the US still maintained an international gold standard. But this too was fractional and had more promises than assets. Countries (most notably) the French did not like how "overbooked" the dollar was to gold and started redeeming their gold in mass. There was a run by international central banks on the dollar and Nixon had no choice but to take us off the international gold standard as well.
So it is confusing, but we went off the gold standard twice. Once to the public (FDR) and once to international central banks (Nixon).
So how did this work before 1972? Gold was the international reserve currency, but on top of this the dollar was the means by which gold would be commonly traded between countries. It was much more difficult to clear transactions in gold than paper (or accounting) dollars, which led to its popularity as a reserve currency. On occasion of course individuals would demand gold for dollars and you would have physical transfers take place (as in load the gold onto the ship and send it over the high seas).
If you are interested in the history of the American gold standard, I highly recommend the book A History of Money and Banking in the United States:
The Colonial Era to World War II by Murray N. Rothbard:
So what is the difference between the old system and the current system? Not much as we were never on a true gold standard. But in a nutshell, banks used to be able to redeem dollar bills or central bank deposits for gold. Now dollars are not redeemdable for anything. You can see on the Fed balance sheet a holdover for this as gold is still considered an asset and dollars (paper and electronic) as liabilities.
So how would trade occur pre-Bretton Woods? Mostly through banks. The receiver bank would obtain credit for gold, or a national currency that was convertible to gold (dollars or say pounds). If one party wanted to convert currencies they could but it mostly took place as a transfer from say "pound deposits" to "dollar deposits". If the new party demanded their payment in gold they could obtain this but it rarely happened.
After Bretton Woods, international trade was mostly facilitated by dollar deposits with the promise of gold redemption of a foreign central bank demanded it (which Nixon ended).
Now when gold reserves were physically demanded, did this cause problems? Yes! Remember, all countries and banks issued more notes/deposits for dollars than they had (somewhat dishonest but that is another matter). Say the pound (during the standard) lost a bunch of reserves...so they go from a ratio of say 1000 central bank deposits to 200 gold to 900 central bank deposits and 100 gold. A serious blow to credibility which could lead to a run on the currency. Do they have to print or destroy more notes/deposits...no because they do not 100% back gold.
So why do countries hold reserve currencies today? They kind of don't need to, but there is a reason. Basically private banks operate by maturity mismatching (balancing short term debt against long term assets). When a currency is not stable (like from a small nation) short term interest rates are too volatile for banking and in fact bank runs can be initiated from clever predator foreign inverters (like George Soros). Politicians like banks...and for some reason they think they are important to the economy, so go to great ends to prop them up. For a small nation this means their currency must be somewhat stable for their banking system to survive...and the nation achieves this by maintaining a peg. You can only maintain a peg if you can manipulate the market, so countries hoard foreign assets like dollars so they can buy and sell their home currency (like say the Peso) so it achieve its peg rate.
As for clearing of international transactions this is mostly done with bank money, but dollars are a common facilitating base for these transactions (although certainly many other currencies are being used now).