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Pixels Per Inch (PPI) and Dots Per Inch (DPI) are two terms that have no DIRECT relation to one another. They are more like second cousins twice removed. :-)
PPI is how many pixels of the image are in one linear inch.
DPI is a measure of how many DOTS a printer has to lay down in one linear inch. A printer has to spit out multiple "dots" to mix colors to render the correct color for the print.
The photographer controls the PPI dimension via the Image Size dialog in the camera menu. If your image is 2000x3000 pixels and you set the PPI to 300PPI, your image will print 6.6x10-inches. HOWEVER, if your printer is printing at 1440DPI, it will lay out as many as 1440 dots in every one of those 10 inches. If your image is 240PPI, your image will now be 12.5-inches wide but the printer will still lay down 1440 dots in every inch of the 12 inches. Depending on the size of the print . the material on which it is printed, and the viewing distance, you can get away with PPI dimensions as low as 150PPI (on a large canvas, for example In some cases, even lower. Generally, 240 to 360PPI is considered normal for prints viewed up close. Different printers have different "native" PPI numbers. (That is another topic for another day and the visual difference is miniscule.)
Printers have to mix the "dots" to make the colors. Unless the color in your images is the exact same as the (for example) magenta in your printer ink, the printer driver has to figure out what colors to mix to give you the exact shade of magenta color needed. And of course there are no brown inks so the printer driver has to figure out the correct mix of black, magenta, yellow and whatever else is needed to come up with brown.