The central bank of a country is responsible of the printing of new currency notes and coins/ money to the economy of a country. My question is how they introduce this new money into the economy in a 'fair' manner. Giving it to one individual or group would be privileging them against other such groups, right? So, how is it done?
One way to think about this is that money is put in circulation when the central bank buys assets from the citizens of a country. Sometimes this is really the main thing going on, for example, to perform QE the Fed bought all kinds of securities from financial markets. When the fed buys these securities it pays banks by increasing the deposit balances of those agents selling the asset.
Another simple way that a central bank "buys assets" is when it gives a loan to a commercial bank. In the process of given the loan, the central bank will give physical cash to the commercial bank and keep the loan in its books as an asset.
This could also happen in an accounting sense but not in an economically meaningful sense, for example when a central bank "buys" bonds issued by the government of its country. The government might never pay or the bonds might loose most of their value if inflation increases interest rates for example, so its a kind of fake purchase.
Finally, economists talk about a 'helicopter drop', where the central bank would print money and then just through it out fro a helicopter, hoping it is evenly distributed among the population... but that's mostly just an imaginary scenario.
[BUT, NOTE: This is a caricature of how physical currency can be printed by a bank and shipped out. Quantitatively, an important part of money creating takes place within the banking system, as banks receive deposits from clients and give out loans, physical money gets "multiplied".]