Some commercial printing companies will stipulate that you need a specific ppi (usually 300) when submitting files for printing. But in reality it is only the total number of pixels which is important, as previously mentioned.
Also bear this in mind if doing your own printing; but most printers will operate with images at 200 ppi with no problems. Your home printer should resize anyway, but for best quality results it can be useful to get somewhere close to the preferred figure.
Increasing the ppi will indeed produce a smaller image but there will be no improvement in quality over 300 ppi, or thereabouts.
In fact, I have a desktop publishing programme which automatically does this and it can be irritating when they end up with something like 500 ppi instead of reducing the image dimensions and still keeping to 300 ppi.
It also reduces the ppi for no apparent reason after I have correctly resized with my photo editor. So when I try to import an image into a text document, and the image is exactly what I want, the software auto resizes it to much bigger and often with an excessively reduced ppi which isn't what I wanted. So I then have to go into that programmes size/resolution menu and resize it back to my original dimensions.
I suppose that is modern technology for you!