Detangling DPI

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Let's talk about DPI.
I've heard people ask, "If I make an image in Photoshop and I want to bring it into After Effects, what DPI should it be to be good quality?"
Or on Deviantart: "I made my canvas 2500x3000 pixels @300DPI."
No. That's not how it works xD

DISCLAIMER: This guide is targeted mainly at people who work digitally, for digital/web output, set their document size via pixels, and rarely print, but still set their DPI thinking it changes something about the quality of their digital pixels. Obviously if you work specifically for print, DPI is going to matter a lot more to you than pixels. In that case this guide is not really meant for you, but the information still applies nonetheless.

Advanced cropping, resizing, resampling in Photoshop from Adobe, in case you still don't understand and/or want to read about more advanced specifics regarding image resolution.

OTHER CLARIFICATIONS:
PPI = Pixels Per Inch

PPI is often used interchangeably with DPI. They are not, however, the same thing. PPI is device-dependent while DPI is image-dependent. (Though if you want to get technical about it, PPI is for screens (Screen Resolution), DPI is for printers (Print Resolution), and the image resolution is neither DPI nor PPI, it's the 'Image Resolution,' measured in pixels-per-inch. The DPI and Image Resolution are so closely related though that they may as well be the same thing, I refer to them as the same thing, and for the sake of this guide I am also referring to them as the same thing, since most people know it as DPI anyway). I know, crazy right?

PPI is for screens. A screen with a higher PPI will fit more pixels into every inch of its screen size, giving you higher quality visuals. The higher the PPI of a screen, the smaller the pixels get, and newer monitors tend to have higher PPI (retina displays, anyone?). It’s why, when you got that shiny new Windows computer, your icons/windows/everything all seemed to be smaller than the ones from your old one.They show you that same 256x256 image in their new, smaller pixels, resulting in a higher quality, less pixelated look. More info about Pixel Dimensions, Image Resolution, and Screen Resolution at Adobe here.

Also, the Resampling section applies only to bitmap images. Resampling images (ie changing their pixel dimensions) is okay for vector graphics, since they're scalable essentially indefinitely without losing quality. But you already knew that, right? ;]

EDIT: Adjusted the wording in a couple of places in the hopes of making it a bit clearer.
EDIT2: Added links for more info, adjusted more wording.

_{Rosie the Riveter painting by J. Howard Miller, retrieved from the website of the Virginia Historical Society. This work is in the public domain in that it was published in the United States between 1923 and 1977 and without a copyright notice. More info here and here}