Back-Focus
Back Focus, or back-focus, is the error of having a subject be accidentally out of focus, and sharp focus instead falling behind or past the subject. While images can exhibit this error even when manually focused, generally speaking a "back-focus problem" is usually referring to when the error is caused by your camera and lens' autofocus.

Back-Focus Technical Definition

The issue of back-focus is commonly seen if a fast aperture lens is used. Sometimes it is the photographer's own fault, however: ether they accidentally focused on the background due to imprecise placement of the selected focus point, or there was actual movement in the scene. One of the most common user-error reasons for backfocus is simply that either the photographer our their subject shifted their balance just a little bit, and you were using an f/1.4 lens standing just a few feet away from each other.

Although technically any image which has focus accidentally falling behind or past the intended subject could be considered to be back-focused, when most photographers talk about back-focus issues with a particular camera or lens, they are referring to a specific problem with the autofocus system that causes the lens to repeatedly miss focus in that "back" direction.

This is an issue that can be corrected with most modern cameras' autofocus calibration or focus micro-adjustment menu function, when performed correctly under controlled conditions.

Back Focus Distance

NOTE: The term "back focus distance", although similar, is an unrelated photography term. It refers to the measured distance between a camera's image sensor plane and the rearmost glass element in the attached lens.

Back-Button Focus

Also note that the term "back-button focus refers to something else too: using a button on the back of your camera to perform autofocus, instead of your shutter release. This is a camera customization that is popular among serious photographers who want to be able to perform the functions of focusing and firing the shutter independently of each other.