One well known example of this technique is the GTA load screen.
Also, I would appreciate it if you could direct me to a good tutorial video as well as the ideal software to use. Thanks!
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You're asking about the fake parallax effect. In the link you posted, it appears that the background is static, but the foreground model is slightly shifting.
For generated graphics, this is trivial: the foreground model is a separate image/overlay that is slowly shifted relative to a background.
For existing static images or photographs, the same concept applies, but with a bit extra preparation work. The model is masked or "cut out" of the image, and enlarged by a slight amount. The enlargement allows for the model / foreground object to be placed where it originally was in ("on", "over"?) the photo, but slightly shifted, creating the illusion of depth by hinting at perspective motion (i.e., implying a subtle, slowly shifting point-of-view of the viewer). Here's a test example I found on Youtube (not my creation).
The fake parallax effect is part of the Ken Burns effect toolbox. Sometimes he just slowly pans images, but often he will employ the fake parallax effect. Software that has a "Ken Burns Effect" built in just uses the slow pan.
Although what you're seeing in the video link is actually an animation, I think I know what you're referring to: artificial motion blur.
In Photoshop you can use the smudge tool, a custom brush tool, or the motion blur filter to achieve something like this in different ways. The Photoshop (or Photoshop Elements) help files are excellent and should tell you everything you need to know about these methods, and with a bit of practice you'll figure out how to get the look you're after. If you're using GIMP or other editing software, you'll have to rely on Google for tutorials specific to the app.