Banks "lend" in a way you and I can't. When a bank lends it creates a deposit "out of nothing" with which the borrower typically purchases something. I will leave aside collateral and mortgages, they are not relevant here.
So by lending, the bank creates an additional liability, the deposit, which the borrower may withdraw at any moment.
The bank has an additional asset which offsets the liability: the asset is the loan. It is an asset because it earns the bank interest fees.
So the bank has increased the size of it balance sheet.
It purports to be able to hand over cash when the borrower withdraws it from their account, but in fact it only has a fraction of the cash on hand. Hence "fractional reserve banking".
Saying "fractional reserve banking" is fraudulent and can lead to bank runs isn't quite getting to the heart of the problem.
The problem is mismatched maturities. To avoid insolvency, a normal business should have short term assets (eg cash) that can be used to pay off short term liabilities. ie they have matched the maturities of their assets and liabilities.
Banks cannot do this. They cannot hope to liquidate the assets (loans) quickly enough to pay the liabilities (deposits). Of course, they could offer 30 year term deposits to match 30 year home loans, but they wouldn't have any takers.
The only reason they are solvent is because the central bank stands as lender of last resort. ie a banking license is a "license to be insolvent".
This is why banks, not mining companies, should be subject to a super profits tax. They could not run their insolvent companies without central bank support.
Or else we could get rid of central banks altogether.
So commercial banks do not "print" money. I know in the UK, some still issue their own notes. But it is beside the point. Private banks basically just leverage up cash so it is utilised more.
So to conclude, private banks increase the effective money supply by private "lending" (balance sheet expansion) and central banks increase it by literally creating money out of nothing.