I think what you are describing is what's called "selective focus." This is where the subject in the foreground is sharply in focus, but the background is softened or blurred out. They do use this technique in magazines and movies a lot, and the reason is to draw more attention to the subject in the foreground.
You would use this technique to blur out a background in a picture that might be too distracting. Let's say you are taking a photo of an insect in a field of wildflowers, or if you want to focus on one flower in the foreground. You would want to blur out the background of the picture because otherwise it would be too distracting.
It also adds a three-dimensional feel to pictures. You would achieve this effect by using a shallow depth of field.
Really, the best way to do this is with your camera. You need a camera that has manual aperture, shutter speed, and focus settings. You would use a wide aperture setting, like maybe f-5.6, which would give a shallow depth of field. Subjects near the camera would be sharp, but the background will be soft and out of focus. But you will also have to compensate with a faster shutter speed in order to get the right exposure...otherwise the picture will be overexposed.
You need either a manual film camera, or a digital camera that has real shutter speed and aperture settings. Unfortunately, most point and shoot digital cameras don't give you that much control and you can't change the aperture setting. It's an optical effect...you have to use a real aperture setting.
You might be able to do this with an image editing program, but it won't look quite as good as getting it with your camera with a true shallow depth of field. Try looking in your editing options and see if there is anything that mimics a shallow depth of field. But really, it's just going to give the "illusion" of selective focus...the best way to do it properly is with the camera.
Is this the effect you were looking for? I took this in April, with a manual film camera. This was with Kodak Gold 200 film. I set the aperture to f-5.6, and the shutter speed to 1/300. Notice how the branch with the greed buds is shaprly in focus, but the background is blurred out.
http://www.geocities.com/gatewaycityca/first_spring_april_16_2008_small.jpg
EDIT: Okay, it looks like I misunderstood your question. Sorry about that. I think DCReed is right, that it probably is an effect from panning the camera while zooming. It might be related to zoom blur, which is another photographic and movie technique. Try looking that up on google.