An HDR image is one image resulting from the merger of two or more images of the same scene, but where the images are exposed differently.
"Pseudo-HDR" is a term applied to one "RAW" image from which areas that are relatively under or overexposed, where HDR software is used in order to recover information lost in the shadows or lost in bright areas.

The degree to which the HDR effect is seen can be controlled so that when "overdone" some of the images appear "fake".
On the other hand, there are scenes where the differences between the brightest and darkest areas (the dynamic range)can't be captured on a single image with existing technology. In these cases, HDR can allow one to see on a single image what they would experience with a system that has a wider dynamic range: their own eyes.