A projector should have a minimum resolution of 800x600, SVGA, for both NTSC and PAL DVD playback. But if you want to use a HD source, or are planning to in the near future, you would be better off to opt for a higher resolution, preferably widescreen. For example when projecting a widescreen, 16:9, movie which has a 720x576 resolution only part of the 600 vertical pixels will be used to project the image, unlike the horizontal resolution where almost all pixels are used. A HD 1080i source however requires at least a native 1280 x 720 resolution projector, and 1080i is all too common in North America.
Projectors are naturally suited to project both interlaced and progressive scan signals. Interlaced signals are comprised of the image divided into even and uneven horizontal scanlines which are displayed one frame after the other, which means that after two frames the full image is formed. Progressive scan signals are generated by combining the two interlaced frames and only sending the full image, not the interlaced frames, which means that processing is needed to calculate the frames in between. With all projectors the signal is always internally converted to a progressive scan signal, which largely determines image quality during playback.
Resolution comparison between two projectors, the left has a 800x600 resolution, the right 1280x720.
The amount of noise the projector produces when switched on is another important factor to consider, especially in a home theatre setup. As a loud projector will be audible over the sounds from the movie you are watching, which never is a good thing as that will draw your attention away from the movie you are watching. Naturally some projector noise is unavoidable as the lamp inside it needs to be properly cooled, thus you cannot simply do away with the fan noise.
Most home theatre projectors have a noise level of about 37dBA which makes them quiet enough for watching movies at a low to moderate sound volume without being distracted. The crux of the matter however is where the projector is located; if you place it right behind you it might still be loud enough to be distracting, so projector placement is very important here. And naturally not everybody thinks fan noise is distracting. Whether or not you find a projector's noise level obtrusive is often a subjective matter which is different for each individual.