In plain English, depth of field is the area in front of, and behind, your subject that is in focus. Your subject can be anything from a mountain to a person's eye.
There is no sharp delineation between a photograph being "in focus" and "out of focus." Instead, a picture will gradually go from sharp to blurry. What causes something to look out of focus is often referred to as the "circle of confusion." This is when a point, whether it's a pixel or a dot on a print, slide or negative, becomes so blurry that the blur can be detected by the human eye. The blurrier the dot is, the more out of focus it looks.
If you're taking a picture of a person, you can have a very shallow depth of field, and only their right eye may be in focus. Or you could have a deep depth of field and not only would their entire body be in focus, but so would the mountains behind them, and most of the ground that's between them and the camera.
It's important to understand depth of field so you can get the photograph you want. Let's say you are photographing a person in a crowded situation, like a Renaissance Faire, and you want them to stand out in the photograph. In this case you would use a shallow depth of field. This would throw the crowd behind him out of focus, and since people are drawn toward objects that are in focus, he would stand out in the photograph. On the other hand, if you are standing on the edge of a meadow, and beyond the meadow is the Rocky Mountains, then you could use a deep depth of field, so both the meadow and mountains would be in focus.
Depth of field is a tool, and you need to learn how to use it because it's in every picture you take, but you can control it to meet your photographic needs.