So, what is a good resolution?

While this is almost the right question, it is dangerously wrong. The proper question is:

What is a good resolution for my specific need?

Resolution is not important for screen graphics; a pixel is a pixel. I use a resolution of 100 dpi as it makes the potential math easier if I am ever trying to use it for printing. Inches make no sense on a screen as monitor size and resolution can be different for every user. However, one inch is 100 pixels is easy to understand when mixing units.

For Printing: I use 300 dpi. Many organizations require 600 dpi. Which is better? Somewhere around 300 dpi, most people will not see the improvement in the graphic quality when printed at higher resolutions—but some will. A graphic with 600 dpi is four times larger than the same graphic at 300 dpi. The question becomes:

When is the increase in apparent quality worth the increase in size?

Modern printers can handle either resolution and even higher resolutions without any problems. The printer is no longer a consideration.

I use 300 dpi because it's good enough for most situations and is a nice balance between quality and space.

However, the real choice must be based on what you are trying to do!

If you are using it on screen, like a web page, size it to the size that looks correct for your design. This is a visual choice.

For printing, you need to know:

1. What width and height, in inches, does the image need to be on the page?
2. What width and height, in pixels, of the original image?

The rest is simple math.