DPI(dots per inch) is often a confusing thing for artists who are starting to try their hand at print. This guide will try to be as simple and non-technical as possible. We’ll start off with an inch, as that’s the easiest way to explain it!
The 72dpi canvas
I’ll be using photoshop for this guide, but I’m hoping most other programs will allow you to input measurements that aren’t pixels. If you open your art program now and create a canvas that is exactly 1” x 1” at 72dpi.

You should have a canvas that is 72 x 72 pixels. This is pretty tiny.

When you print this it end up really pixellated like so: (not to scale)

): You didn’t want this. Now everyone has been telling you that you should make your canvas 300dpi as that’s what you do for print.
Unfortunately if you have already drawn your image at this size you can’t just change the dpi. This will in fact make the printer print your image even smaller.
72dpi vs 300dpi
The easiest way for me to think about this is that every pixel is a “dot” for your printer(sorry to anyone who knows the technical side of this).
You have this 72dpi canvas and you are telling the printer you want this image to have 72 dots/pixels per inch so your printer is printing off these bigger dots to fill that inch. This results in that pixellated look that you are probably familiar with.
You now change the canvas to 300dpi. This canvas now prints out at 0.24” because your printer is now cramming 300 dots/pixels into that inch so each dot has to be smaller. This results in a much clearer picture but now it is tiny.
The 300dpi canvas
Now armed with that knowledge set up a canvas that is 1”x1” at 300dpi.

Your resulting canvas should now be 300 x 300 pixels.

Much bigger! Now if you draw an image on this canvas and print it out it will be a lot crisper and clearer. However it also means you had to draw a much larger picture. This is the unfortunate reality of print. Everything needs to be over 3 times the size of standard screen.
This makes it really important to know your physical print dimensions as it is much easier to create an appropriately sized canvas using inches, millimetres and centimetres(I’ll get to mm and cm in a bit as well) rather than guessing what it could be in pixels.
Centimetres & millimetres
Since I’m Australian myself we don’t use inches, which initially left me a little confused. How does dpi work for us? Well it pretty much works the same way.
There unfortunately isn’t a dots per millimetre that I know of, but you set up your canvas using mm or cm and 300dpi and your program will be able to work that out for you still. The thing is that now after you created your canvas and you check the dimensions it will often has decimal points.
This is such a small measurement it won’t effect your printing. It’s basically because programs can’t create a canvas that has a half pixel or third of a pixel. So it will give or take a pixel to try and get it as close as possible to your cm/mm measurement.
A comparison
This is sort of a little bonus/sometimes if you think your canvas is too huge for print. It probably isn’t. Here are some common sizes and how big they print at 300dpi.
800 x 600px - 2.667 x 2in - 6.77 x 5.08cm - 67.73 x 50.8 mm
1024 x 768px - 3.413 x 2.56in - 8.67 x 6.5cm - 86.7 x 65.02mm
1600 x 1200px - 5.333 x 4in - 13.55 x 10.16cm - 135.47 x 101.6mm
As you can see these probably look huge on you screen but none of them are even close to an A4 page. Here is the reverse. I’ll show some international paper sizes with their 300dpi pixel sizes!
A5 - 148 x 210mm - 5.827 x 8.267in - 1748 x 2480px
A4 - 210 x 297mm - 8.267 x 11.693in - 2480 x 3508px
A3 - 297 x 420mm - 11.693 x 16.537in - 3508 x 4961px
As you can see these are rather large. So if you ever create a canvas and think you’ve done something wrong because the canvas is freaking massive. You are probably doing it right haha.
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Hope that makes any sense!