Printing money does not always lead to inflation. To understand why not first we need to know why it may cause inflation. I will try to explain without any graphs or equations.
Printing money is kind of a trick, the government is trying to trick you into believing that more money means more income even if the real output remains the same. The weird thing is that it works sometimes. It works because the economy is a system which has frictions in it and does not adjust everything instantaneously.
(i) More money
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(ii) More demand for treasury bonds (this is generated by the FED buying a lot of treasury bonds from the bond market with the money they print)
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(iii) Bond prices rise
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(iv) As bonds have fixed return on maturity, a rise in their price means that the difference between their end value and their initial value decreases, as a result their yield decreases which means the interest rate of the bond falls
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(v) As bond interest rate falls lenders now try to replace government bonds in their portfolio with other form of assets. The only way to do that is to lend out more to other borrowers. But to get more borrowers they will have to attract them by offering cheaper loans, that is at a lower interest rate.
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(vi) Interest rate in the overall market falls
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(vii) Investment rises, as it is now cheaper to borrow money and so more profitable to invest
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(viii) Rise in investment generates employment.
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(ix) In labor market now there is high demand for workers.
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(x) Wages start to rise in the overall economy.
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(xi) As wages rise, it becomes costlier to produce goods, not only investment goods but all kind of goods.
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(xii) To keep their profitability companies raise their price of products.
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(xiii) Inflation!
If there is little unemployment in the economy then output does not rise at all. Only workers shift from their existing jobs to jobs which has undertaken investment for expansion as they are now offered a higher salary. So the wages rise but at the same time prices have risen too. As a result real wage remains the same. We end up where we began except for the fact that we now have a higher price.
Now the reason why it works sometimes without generating inflation is when there is high unemployment already present in the economy due to recession. If that is the case then the step (x) will not necessarily occur. Employment generation will only take in the surplus labor who are unemployed without a rise in wages in the economy. If that happens we will have higher employment and output with little impact on inflation.
This is the most basic model. There are other more nuanced reply to this question. For example Ricardian Equivalence approach says that treasury bonds are not part of real assets and a rise in their price or fall in their yield will not have any impact on the portfolio of assets. But that is all very complicated to fit in one answer. This is the most basic undergrad model.
