The US Government sells treasuries and bonds to raise dollars. These have to be approved by congress, which is why every so often you'll hear about government shutdowns, because, yes, if for whatever reason, the US Congress doesn't pass a debt ceiling lift, the US Treasury will not have enough money to run things anymore.

Then there's the Federal Reserve, which loans dollars to banks to create liquidity and velocity in the monetary system depending on certain conditions (Primarily inflation and unemployment I believe).

(Presumably there's a fair amount more to it than that as well).