Video up-conversion is the science of using a computer or processor to add resolution to a video signal. The first efforts in this space were made popular by the Faroudja LD-100 line doubler, which was an analog video processor that could de-interlace 480i NTSC video scan lines to offer consumers their first look at progressive video. The effect at the time - the early 1990s - was quite impressive, as was the cost.
With the advent of HDTVs, upconverting, also called "scaling" was a requirement instead of a luxury. To display a standard definition signal (480i or 480p) on a 720p or 1080p TV, the image needs to be scaled to fit the screen. How well the scaler does this is a major factor in the performance difference between different AV products.
Internal video processors that scale and deinterlace are inside DVD and Blu-ray players and even receivers and AV preamps. All HDTVs and projectors have internal scalers/deinterlacers as well. Some companies, like Anchor Bay, still make external video processors.
Converting 1080i (what you'd get from your cable/satellite box) to 1080p is not technically upconversion, but de-interlacing.
It is important to note that some purveyors of video up-conversion claim it can turn a standard-definition image (480i) into a perfect 1080p video signal. However, an unconverted image is not the same as a real, native HD signal. No upconverted image will look as good as a Blu-ray.