One question I get asked quite a bit on my Lightroom course is "what is a good resolution to print?" The Lightroom export module allows you to choose many different options when outputting a file for single use. But before we look at resolutions, lets look at what's going to be looking at your images - the human eye.
Assuming 20/20 vision, the average eye can resolve 876ppi at about 4 inches. This is the closest detail resolution you can see if you really scrutinise an image up close. As we move back further away from an image, that number goes down - and it goes down quite dramatically. For example:
At a standard reading distance (for example 12") the eye can only resolve about 300 ppi.
At a standard monitor distance (for example 30") the eye can resolve about 115ppi.
At a TV viewing distance (for example 6 feet) the eye can resolve no better than 50ppi.
Looking at a cinema screen distance (about 40 feet) the eye can't do any better than 7ppi.
So viewing distance is a huge factor in determining how big you can go. Let's look at the print itself.
An image is (very basically) made up of pixels - the number of which you can work out from your camera sensor specs. My Fuji X-T1 sensor has a resolution of 4896x3264.
Dividing the longest edge of the sensor (4896) by 300 gives me 16" on the longest edge as the biggest I can print with superb - up close - quality.
Most of my prints go on walls though, so with a viewing distance of 30" assumed, my 4896 sensor will now print to (4896/115) a whopping 42" on the longest side (115 ppi).
So without any sizing, it's quite easy to determine what resolution is "best" for you.
Resolution (pixels per inch) = sensor size (pixels on longest edge) / size (inches)
So if you want to print to 8 feet with your X-T1 the resolution will be 4896 / 96 or 51ppi - perfectly acceptable for viewing at 6 feet and beyond.
If you wanted a billboard that could be viewed at 40 feet without loss of detail, you could print to a huge (7 = 4896/x) 58 feet along the longest side!
We can go even bigger though using resizing algorithms. The method above will result in your software neither adding or subtracting pixels from your image. Going smaller is quite straightforward as Lightroom is excellent at downsizing. Using the examples above, when I print an A4 image from my X-T1, LR tells me that my ppi is 463 (10.5" image size with small border divided into 4896 pixels).
Going up though is a bit more hit and miss. Lets say I wanted to make a 20" print. 4896/20 = 244ppi, but if I wanted better resolution (say 300ppi), I'd need to "add" 56ppi in order to get that. The software would need to artificially create pixels to compensate. This isn't something I want to do and in this case, I'd feel that 244ppi is an acceptable print resolution for an image that big.
Monitors, TVs and tablets (and phones) are slightly different. They already have a "size" in pixels which may be independent of the size of the screen. My 38" TV has a 1024x768 resolution whereas my 24" monitor has 2560x1440. So it's possible to have more pixels in a smaller space (the iPhone 7 has a 760 x 1334 display in a 4" screen which is better than my TV!) Because of that...
For screen display, "inches" are irrelevant.
As an example, a 1000px image would take up my entire TV screen (30-odd inches) but that same image would only take up half of my PC monitor (12"). When exporting an image for use on a screen, consider only the number of pixels in it. In the Lightroom export module, that means ignore the DPI box.
Most browsers will resize an image to fit, so it can be tempting to upload full resolution images. However these can easily be downloaded and printed out by someone else so what resolution should you upload to avoid people being able to print?
Well, if we assume that people want a decent image quality of at least 200ppi we can do some more maths.
If you export your images at 500px they will look tiny on a high resolution monitor (they'll fill a quarter of my screen) but they will only print out to (500/200) 2.5" on the longest side. That's pretty small and useless.
If you export your images at 1000px they will look ok on a high resolution monitor (they'll fill a half of my screen) but they will only print out to (1000/200) 5" on the longest side. That's quite a big difference.
If you export your images at 2000px they will look great on a high resolution monitor (they'll fill almost all of my screen) but they will print out to (2000/200) 10" on the longest side. That's a decent free print for someone. Even at magazine quality 300dpi, that's a 6.6" print!
So the upshot of this is that there's no definitive answer. It depends on what you're trying to do with your images.
In summary, most modern cameras today are capable of excellent quality images printable to any appropriate size for the viewing distance. Even my monitor (2560x1440) is even capable of getting game screenshots that will print out brilliantly to A4. And that's effectively only a 3.6MP camera!
Dots vs Pixels
There's an ongoing debate about ppi vs dpi. Dpi generally refers to dots per inch and is a printing term that relates to how many dots a printer can lay down per inch. When looking at digital images and thinking about how they print, dpi isn't really a concern. As a photographer, all you should be concerned about is pixels per inch (ppi).
The unfortunate problem is that some printers will interchange dpi & ppi freely - whilst meaning the same thing. A friend of mine was recently asked to supply an image "at 300dpi" for example.
1. You can't supply an electronic image that references dots
2. Any image is 300ppi. Even a 600 x 400. it's just that it will only print to 2x1.3"
You need to also know what size you want to print to before you can supply the correct image. For example, an A4 (8"x11") print at 300ppi would need to be at least 3300x2400 to qualify