Intagio printing is the raised print on a banknote. It is high quality, expensive, and difficult to forge easily or cheaply. Intaglio printing is usually the final print run on a banknote. This printing is done from an engraved plate, where the design is cut out from the metal and the ink is applied from these cut areas.
A less expensive high quality printing typically on banknotes is offset printing, which prints the flat colour on banknotes, usually in one, two, or three runs, depending on the level of security and face value of the note:
https://www.fleur-de-coin.com/eurocoins/banknote-production
https://www.komori-currency.com/
https://www.jenningsprint.com.au/how-does-offset-printing-work/
Letterpress printing is an older printing process, often used to add the serial numbers. The process involves using a raised surface to apply the ink.
Screen printing is not used on banknotes. It uses a mesh to apply ink. The best resolution from this process is around 240 dots per inch (dpi).