How do you actually do all this in Photoshop?
The 'Image Size…' dialogue in Photoshop – or similar apps – looks simple, but it's a minefield if you don't know what you're doing. The key thing to remember is that all the fields are essentially connected (because the pixel dimensions, resolution and physical printed size are). Changing one value will change others too, depending on what kind of resizing you're trying to do. Here's a guided tour.
Aspect ratio (1)
This maintains the aspect ratio – the relationship between the width and the height. For almost every job you do, this should be active. Click to toggle.
Units of measurement (2)
You can see the image size displayed in pixels, centimetres or other measurements. Note that of course everything's still interrelated, so if you switch to centimetres and change the values, the pixel count might change as well.
This is the DPI (or, more correctly, the PPI; clarification coming…) of the image.
Resample option (4)
This is the tricky one. With this Resample option unchecked, the pixel dimensions (the number of pixels across and down) of your image won't change as you tweak the values. You're essentially just making each individual pixel bigger and smaller for output (as in Example A on the previous page) when you adjust the resolution or physical size. With this box checked, you're changing the pixel dimensions, either up or down. If you ultimately add more pixels (keep an eye on the Dimensions readout at the top), they'll be added using interpolation (Example C).
Resample types (5)
You have some options for how the image is resampled here; if you're unsure, leave it on Automatic. 'Nearest Neighbor' is the only option not to use interpolation, and you should usually not use it; a common exception is if you're dealing with pixel-perfect graphics such as low-res screenshots or pixel art, and you want the detail not to get smeary when you boost or reduce the resolution. In this case, work in increments of 100%.
That should hopefully have given you a basic understanding of the role of DPI and resolution, but I bet you still have questions. Let me guess what they are and try to answer them!
What DPI should my images be at for print?
For print, 300dpi is usually optimal, but you can get away with less; even 150dpi is often okay for photographic images, though it would be starting to look a bit rubbish for logos and the like. However…
What DPI do I need for screen?
Most people are under the impression all images for screen should be at 72 or 96dpi. However, DPI is broadly irrelevant for designing images for screen, and not just because 'DPI' is specifically a print concern. Because you'll be presenting a grid of pixels (your image) slotted one by one into another grid of pixels (a screen), 'the size of the pixels', which is what we're talking about when we're talking about DPI, is irrelevant. So in general terms, you just need to care about the pixel dimensions, not the DPI/PPI.
The 72/96dpi thing is a legacy kludge, and you can mostly ignore it, especially with today's high-resolution displays, which have their own problems with creating @2x versions.
There's a wrinkle, though: in some contexts, systems can try to read a DPI value from metadata and be clever about how they present an image, when all you want them to do is just display the pixels 1:1. So, much as it annoys me, I do sometimes manually set the resolution of images destined for web pages to 72dpi.
Why does my printer print at 4800dpi?
Each pixel on a normal modern computer screen can display one of 16.8 million colours, but when that pixel is printed on, say, an inkjet printer, unless it's pure cyan, magenta, yellow or black, the printer has to recreate what colour it is by placing even tinier dots of pure C, M, Y or K ink within the bounds of that pixel so that our eyes perceive the pixel as the correct colour.
Therefore, the number of actual dots per inch the printer needs to be able to place has to be high – much higher than most of the images you'd send to it – in order that you can create the colours necessary for each pixel. And that's why conflating DPI, PPI and SPI as I did early on is actually a terrible idea, technically.
Many people do just call all three 'DPI', however – even if Photoshop does correctly call it PPI in its dialogue box – and unless you're getting technical or pedantic, you usually can too. (There's LPI too, but let's not complicate things further.)
What is image 'resolution'?
At this point, you're probably wondering, when someone asks for the 'image resolution', if they're referring to the pixel dimensions or the DPI (or PPI)? Well, it can be either. Sorry. Different people use the term for each, and anyway it depends on context. If you're unsure, clarify.