The reason for this is resolution. That is, the number of pixels per inch. Artwork that is created for the web is low-resolution, artwork that is created for print is high resolution. And if you have spent a lot of time on a project in Photoshop, without giving any thought to resolution, you may be heading for trouble. Of course, you could use a vector program like Adobe Illustrator, and never have to think about any of this, but Illustrator doesn't always work for all types of artwork effects.
So, before you launch Photoshop, determine the size that the image will need to be. Most web pages are no bigger than 1200 pixels wide, so there is really no reason to make the artwork any bigger than that. Again, be absolutely, positively, sure that the artwork will not be needed for print. That is a completely different issue, and one that I won't address right now.
I recently designed a piece of art for a client that they will use on their web page. When I met with the client, I made this point absolutely clear. If the artwork was going to be used for both web and print, the price would have been considerably higher. Designing for web is much easier than designer for print
This is how I did my most recent project -
Based on the photos that were supplied to me, I worked on an canvas size that was a little larger than ten inches wide (720 pixels). Actually I went to 900 pixels. Now, the artwork will never, ever, be displayed that large on page, so when I save it for web it will only get better, and sharper, as it gets smaller. It will probably be used at no more than 400 pixels wide, and I will get the exact specs from their webmaster when the artwork is approved.
From a technical standpoint, this is a very safe approach. From a design standpoint, don't forget to view your artwork at a smaller size. Fine lines, tiny type, etc. will disappear. In addition to thinking about pixels, be sure to think about design!