Definition: Aspect ratio is the relationship of an image's width to height, or its proportions. Digital cameras produce files with an aspect ratio of 4:3. But many common photo print sizes have a different aspect ratio. For example, a 4 by 6 print has an aspect ratio of 3:2; an 8 by 10 has an aspect ratio of 5:4.

When you change the print size of an image and you're going to a different aspect ratio, make sure the values you enter will accommodate the whole picture, otherwise your print will have gaps along two sides. Of course, the trade off is that a small part of your image will be clipped.

Take, for example, a typical digital camera file that you want to print at 4 by 6 inches. With aspect ratio restricted, you can either do 6 by 4.5, resulting in a half inch of the height being clipped. Or you could do 4 by 5.333, resulting in a blank strip along the top and bottom edge of the print.

To avoid having the image arbitrarily clipped by your software or photo printing service, you should crop the photo to the correct aspect ratio, prior to setting the print size. Many newer software will have preset aspect ratios in the crop tool for common photo print sizes. In Photoshop and Photoshop Elements, you can enter the height and width in the options bar before making a crop selection to crop to a specific aspect ratio--just avoid putting a number in the resolution field if you don't want the image resampled when you crop it.

Source: Aspect Ratio and How to Crop to the Proper Print Proportions

That's a good definition and I just saved myself a bunch of typing. Now here's why it's important when shooting for a client.

First, I do my own custom printing. But, even if I didn't, as a photog, I have to be able to offer my clients different sized prints i.e. 4X6, 5X7, 11X14, etc. 

This is where ratios come into play. I was always taught to visualize the shot before you take it and compose the shot in the viewfinder before you squeeze the button. My camera, like most, shoots at a 3:2 ratio. If I take the picture exactly the way I compose it, what happens if the client wants an 8X10 print that uses a 5:4 ratio? I'm going to lose some of my composition.

Here's an example where I composed the photo exactly as I wanted it in the viewfinder. Trying to crop it for an 8X10 print doesn't work.

The 3:2 ratio on the left is exactly as I composed it in the camera. The "S" curve reflection in the pond behind the subject is really what makes the picture. This is fine if the client wants a 4X6 or an 8X12. But, if they want an 8X10 (5:4 ratio) the crop on the right is the result. I can either keep the pond reflection and cut off the subjects feet, which would look horrible, or lose part of the reflection in the pond. In doing so, it totally destroys the composition I was after in the first place.

So whats the solution? Add a little bit to your shot. Try to visualize the crop when composing in the camera. When visualizing the crop, try to leave enough space so you can make a few different sized prints out of what the camera captures.

Here's another example. This is the shot as I saw it through the camera:

 It's already a 3:2 ratio, but leaving extra room allows me to crop it to 5:4, and also a new version of 3:2.

 So now I can print an 8X10 if I want to without losing anything I wanted to keep in the picture.

Let me point out also that this is how I choose to shoot. Many photographers will simply leave the composure as it is in the camera. If the client wants a size that is not suitable to the crop, some photographers will simply add a black background to the print. Personally, I like my way better and I think it's a better way to fill the clients needs.