With the caveat that I’m far from an expert on macroeconomics:
As I understand it, high levels of inflation are basically an issue of supply and demand. They occur when supply cannot keep up with demand; this can be a result of demand being stimulated, or of supply being restricted, or of both. The current inflation is due to ‘pent-up’ demand from the pandemic being released, so to speak, at the same time as global supply chain issues and shortages. On the flip side, you can get a deflationary spiral when consumer confidence collapses, most notoriously as in the Great Depression, though this is very rare, but when it does happen, often far more catastrophic than high levels of inflation.
This is why raising interest rates is the main tool available to central bankers to control inflation; it reduces demand by encouraging people to save and disincentivising them from borrowing, leading to them being less willing and able to spend.
Ultimately, this does all link in with the money supply; an increase in demand for instance, can often be seen as the money supply increasing e.g. people being paid more or receiving government stimulus.
The mainstream view in economics is that an increase in demand does not increase the money supply, only the velocity of money.
Only the Federal Reserve can increase the money supply by either printing more money or by lowering reserve ratios.
This view is challenged by the notion that with modern technology, private banks don't need any reserves, but can simply loan/create money out of thin air. The Bank of England refers to this as 'Fountain Pen Money.'
I don't know people being paid more or receiving government stimulus are the main beneficiaries of private banks creating money out of thin air.
The distinction between the 'velocity of money' and the money creation process gets to the difference between money and physical currency. 'Money" itself is placed into further ever broader categories of M1, M2, M3 and M4. All of that is more technical than is required to understand the basic process.
While, again, this is in dispute in the modern era given the roll private banks play, the basic of the money creation process works like this.
Let's say the Federal Resevere places $10,000 into the economy by buying a bond held by a private bank. Since this is new money to the private bank A and since this is money received from a deposit, the private bank is free to loan out the whole $10,000.
The money is loaned out to person X who then goes and buys something with it at store A. Store A receives the money and puts it in Bank B. With a 10% reserve ratio, Bank B can loan out 90% of the $10,000 ($9,000.)
Bank C will eventually loan out 90% of $9,000 or $8,100.
This process plays out, assuming no currency drain (money that is not placed back in a private bank) with $100,000 in new money created from the $10,000 in currency that the Federal Reserve put into the economy by buying the bond from private Bank A.
Leaving out 'currency drain' this process has a simple arithmetic formula of:
Currency printed by Federal Reserve/reserve ratio
in this case: $10,000/0.1 = $100,000
If Reserve Ratios were reduced to 5% instead of 10% the arithmetic formula would simply be:
$10,000/0.05 =$200,000
So, the Federal Reserve, based on mainstream thinking, controls the money supply process. What private banks and citizens control is the 'velocity of money.'
In this case, that is, how long before the $10,000 in new currency that the Federal Reserve
used to by the bond is turned into $100,000. The Federal Reserve can only regulate the 'velocity of money' by raising and lowering its interest rates. (It can also do so by raising and lowering the 'overnight rate' that banks lend at to other banks, but again, that is more technical than need be and doesn't alter the distinction between expanding the 'money supply' and the 'velocity of money.'
