The resolution of your digital image will have much less effect on its appearance than the dimensions. I still wrestle with the difference even though my images have now been projected on darn near every modern kind of screen.
While the resolution, pixel per inch, will make a subtle difference as your display device tries to grab enough pixels to reproduce your original, the dimensions, the number of pixels along each side, will have a more obvious effect on what is viewed.
In the movie below you'll see a dramatic difference between screen grabs of one of my images created with the same dimensions as my computer screen and one with a far smaller dimension. Both have 72 pixels per inch. You'll also note that the same image shown on the huge screens at the convention for which I shot it looks pretty good. I'm told by A/V people that this is because the projection system "translates" the image data to fit the screen.
Though it seems arbitrary, I use 72ppi for my web images because it works. If you're unsure about which resolution will work best in the expected display, it's best to go large. The resolution of the screen will also make a difference so give it plenty with which to work.
Look at the difference between images with the same resolution but different dimensions.