A line doubler takes interlaced video, like NTSC's 480i signal, and de-interlaces it into a progressive scan 480p signal. There are HD line doublers for taking 1080i and converting it into 1080p, but I think you are looking more at NTSC input.
A scaler will convert the video feed from one resolution to another. An example would be feeding 480p into a scaler from a progressive scan DVD player or a line doubler and it sending out 1366x768 to your projector.
There has been a lot of work towards consolidating these two different things into one box that takes 480i in from a source and de-interlaces it, scales it to another resolution and outputs it to a device without having to convert from a digital signal to analog and then analog to digital again.
One other thing that is sometimes done is frame rate conversion. Typical video displays are running at 60Hz (refresh rate), but film is shot in 24 fps. To make the film stay in sync with video, they play the first frame 3 times and then the next one twice. This causes panning shots to look jerky. If you have a display device that can run different refresh rates, you could set it to be at 72Hz, thereby playing each frame 3 times and getting rid of the jerky look in pans (called judder).
A simple home theater PC can do all this very cheaply. A well built HTPC can easily outperform stand alone systems costing five-ten as much.