DPI vs Print Size Quality
The way I try to explain these relationships, and it can get confusing, is to think of an image having a finite number of pixels in it.
Let's say the native image out of the camera is 900px high x 1200px wide at a dpi of 300.
That means that there is 300 dots of ink per linear inch, giving a print size of 3" x 4" (900/300=3, 1200/300=4).
Let's say in PS under Image Size, ensuring the "resample" box is unchecked, you change the value in the dpi box to 100 dpi. What essentially happens is that PS is being told to spread out the pixels in the image, so that there is now only a 100 dots of ink per linear inch. So the print size is now 9" x 12" at 100 dpi.
900px/100dpi=9", 1200/100dpi=12"). There is still the same number of pixels within the image, they have just been spread around. Thus the image at 3"x 4" at 300 dpi and the 9"x12" at 100 dpi are exactly the same file size.
The issue comes in when you check the resample box. That tells photoshop to either add or discard pixels independent of the dpi.
I like to think of the resample box as the "adding or throwing pixels away" box.
Example - take our original 900 x 1200 pixels image at 300dpi at 3" x 4".
If a magazine wanted you to send this as a 6" x 8" image at 300dpi, you don't have enough pixels in the file. Changing the size in PS in the Image size dialog box would result in a lower dpi since the pixels need to spread over twice the area. This results in a size of 6" x 8" at a dpi of 150dpi.
Since that is below what is needed for a nice magazine print (300dpi), then you would check the resample box, and change the dpi from 150 to 300dpi. You will see that the size and dpi are now independent of each other and that by checking the resample box, you are essentially telling Photoshop to make a guess and add in the extra pixels needed. This is known as interpolating up, or rezzing up.
The opposite, when asked for a file that is smaller than the native size results in a dpi higher than 300, you do the same resampling to discard the extra information and reduce the file size.
So, dpi and pixels vs print size can be confusing depending on what the output is. Hope this helps and isn't at all patronizing?
best,
julian