In the world of digital images, dots/pixels are the fundamental measure of file size. Read the following for a general overview or skip ahead to a specific topic:
"dpi" stands for "dots per inch," and it refers to the dots of ink that a printer applies to paper. High-quality printed images are 300 dpi or higher—the dots are so small that the naked eye can't easily distinguish them.
The lower the dpi, or print resolution, the more pixelated the printout will look, since you will be able to see the individual dots.
Your monitor screen is a gridded array of pixels—tiny squares of light. Most monitors display about 72 pixels per inch.
A digital image can be described as X pixels by Y pixels. Changing an image's resolution, technically, does not involve actual gain or loss of information. All it means is that you are expanding or compressing the dots/pixels.
You can increase the resolution of an image by squeezing its pixels closer together, which makes the picture look better when printed out, although its physical dimensions on the page will be correspondingly smaller.
You can also expand the pixels, thus lowering the resolution, which makes the picture print out bigger. But as the resolution sinks below 300 dpi, your printed-out image will look more and more pixelated. High-quality museum books require images that are 300 dpi or higher. Magazines require images that are 240 dpi or higher. The typical computer monitor displays at 72 dpi, so any higher resolution is not useful.
A file that is 300 x 300 pixels can be:
3 x 3 inches at 100 dpi
1 x 1 inch at 300 dpi
If you scan a 4 x 5 inch transparency at 300 dpi, you get a digital image that is 1200 x 1500 pixels.
If you scan a 1 x 1 1/2 inch slide at 1200 dpi, you get a file that is 1200 x 1800 pixels.
From this slide scan, for instance, the 1200 x 1800 pixel file will look good on the computer screen at up to 16 x 24 inches, since the screen is 72 dpi. It will look good in a printed book at up to 4 x 6 inches, since the printed image is optimal at 300 dpi.
When you reduce the pixel dimensions of an image file (aka downsizing it), you are losing information, but as long as the printed resolution is still 300 dpi or more, you haven't compromised the visual quality of the final printed product. Conversely, you can't increase the pixel dimensions of an image file very much before evidence of visual artificiality will be apparent (images that look washed out, or pixelated). You can't add information that isn't there; photo editing programs such as Photoshop can interpolate only to a very limited degree before an image starts to look bad.
Since you can't bring back information once you've thrown it away, one good safety practice is to save a copy of a new digital file (one you've just received or one you've just scanned) in a separate place before you begin manipulating its size and resolution. This way, if you accidentally hit "save" after downsizing the image, you still have the original, larger version.
All of these operations can be done in Photoshop via the Image / Image Size menu. Be attentive to how the numbers change according to whether the resampling box is checked. Changing the resolution will change the size at which the image prints, but it won't add or delete information as long as the resampling box is not checked.
File formats: JPG vs. TIF
JPG and TIF are different file formats. They are analogous to MP3 and WAV file formats. When you save a file as a JPG (as with an MP3), you compress the data and suffer some data loss. The amount of data loss is configurable. There is no such compromise with TIF files, but the file size is thus typically much larger than a JPG of the same image. For print projects, CMYK TIF files are preferred because image quality is more important than file size, and printing is always a CMYK process. RGB JPGs are used on the web, where file sizes need to be as small as possible for faster downloads.
When you save a file as a JPG in Photoshop, the last thing it asks you is to specify the quality of the JPG on a scale of 1 to 12. This determines the degree to which the file is compressed, and it is a totally separate question from the image's resolution or pixel dimensions.
Saving a file as a JPG always degrades the image quality, but when you save a JPG in the "high-quality" range (10–12) the degradation is generally not noticeable. The lower the quality, the smaller the file size, but the worse the appearance of the image and the greater the data loss.
Again, this kind of quality loss cannot be regained because you are throwing away information. In other words, you can't save an image as a low-quality JPG and then re-save it as a high-quality JPG and expect to get back all of the digital information you discarded during that first save.
Emailing image files
When you email an image file (unless it's for some specific purpose that requires a high-quality image), you should economize on file size. The first step is to save the file as a JPG if it isn't already.
Then check the resolution: If the recipient is only ever going to look at the image on their computer screen (for instance for a Powerpoint demonstration or a website), then there's no reason for it to be at a higher resolution than 72 dpi at the size of a standard computer screen. Most people's screens are 12-22 inches across, so an image that is 6 or 8 inches across (500 pixels wide, in other words :) is probably plenty big.
Providing images for print publications
When you are submitting a digital image for a print publication, you should provide the largest file you have. Do not downsize an image before submitting it, since your designer can always reduce an image, but they can never recover the information that was lost when it was downsized.
TIF format is preferable to JPG format, but if all you have is JPG (for instance if the image was taken with a digital camera and never existed in anything but JPG format), then you should send the JPG, rather than converting to TIF.
The idea is to keep the image in whatever form/format it was in originally, since manipulation and intervention often degrade an image's quality.
If you hire a photographer to shoot new digital files, they should ideally be 10 x 13 inches or larger at 300 dpi. This way the image could theoretically fill up an entire page of a large book. (If you plan to spread the image over two large pages, then the file would need to be twice that big, or 10 x 26 inches at 300 dpi.)