Back-focusing and front-focusing is when the auto-focus consistently is slightly off in either direction.

This problem has always existed as long as auto-focus has existed, but it has come into focus (no pun intended) with digital cameras where you can enlarge the image to pixel level and really see where the focus is.

You can test this with a simple setup of a ruler and a box of matches, or any similar objects. Put the ruler on a table and place the box standing right beside it. Focus on the box and take a picture, preferrably using a long lens and the largest possible aperture (lowest f-stop value).

box -> |                         o <- camera
       |
--------------- <-ruler

Now when you examine the photo you can see where the ruler is sharp. If the auto-focus is correct, the box is in focus, and the part of the ruler that is sharp is a section in front of and behind the box, centered slightly behind the front of the box. That is because the depth of focus is slightly longer behind the subject:

box -> |
       |
--------------- <-ruler
     ^  ^
     |__|sharp