A good printer resolution, specified as Dots Per Inch (dpi), is important to obtain a quality print. Generally the more, hence finer dots per inch, the better the quality of the final printed image. Most modern colour inkjet printers have a resolution of between 600 dpi and 1440 dpi. The latest Epson printers now have resolutions up to 5860 dpi!
However, the resolution of the image sent to the printer driver software does not have to be anywhere near the printer resolution. Here's why. Each individual dot in the image can be made up from a colour palette of usually 16 million colours or more. The printer has to fool the human eye into thinking it has printed this exact colour. Well it cannot, it only has at best five colours, the white of the paper, and black which it can use. It must therefore lay down a grid pattern of dots on the paper (within this limitation of colours) to fool the eye. So, for every dot in the original image, a grid of say 8 x 8 coloured dots is produced on the paper.
Hence it is usual for Epson Photo printers, having a printer resolution of 1440 dpi, to only require an image resolution of 250 to 360 ppi )pixels per inch) at the final print size. Of course the higher the better, but anything higher than the maximum and the printer driver will throw the information away because it cannot handle it. Why have too big a file which the printer cannot usefully use?
Remember that to obtain 300 ppi at A4 size, you will need to scan in the image at 2400 ppi (often incorrectly stated as dpi) from a 35mm negative, to allow for enlarging it eight times. Image resolution can be boosted by using software interpolation, but this is not precise enough for most photographic images and some inaccuracies may result. Hence the optical resolution for any scanner, either flatbed or 35mm, must be at least 2400 pp (or dpi) for satisfactory enlargement to A4 size. You can just about get away with 200 ppi printed image at A3 size, or slightly more ppi with a small amount of interpolation.
The latest Epson Photo printers can also vary the size of individual droplets of ink.
Colour laser printers and solid-ink printers can mix the colours as they are laid down on the paper, so they do not to fool the eye in the same way using a grid, hence these only need to have a resolution of generally 300 to 600 ppi.