Seigniorage is the profit made by a government through the increase in the amount of money in circulation: the monetary base.
New coins minted and banknotes printed cost far less to make than their face value, and the government makes a profit on this. Increases in banks balances with the central bank cost zero.
The government also saves money when newly issued money is used to buy government bonds, reducing interest payments. This is usually the most important source of seignorage profit. This is a regular part of a central bank's open market operations.
Even governments that are printing money to fund expenditure generally do so through the central bank to buying up the otherwise excessive supply of government bonds.
The government does not make a profit on replacing old coins and notes: in that case the new currency is issued to holders who return the old to government — usually commercial banks return obsolete or worn cash to the central bank and receive newly minted replacements. There is small amount of profit from replacing currency that has been completely destroyed, but this is not significant.
Seignorage profits are greatly increased if a currency is widely held abroad, as this money is taken out of the economy and its issue is therefore not inflationary. This requires a larger monetary base.
As one would imagine, this means that the US government gains very large profits from the wide use of its currency in international trade and as a reserve currency (held as part of other countries reserves of foreign currency). The issuers of the other major currencies also benefit to a lesser extent.