IANADC (Digital cinematographer), but I probably know just enough about the subject to be dangerous.
As usual, Wikipedia has a very good entry on the subject.
Your candidates are probably referring to the distinction between 1080 (HDTV) and 2K digital cinema. (They may also be interested in knowing what recording medium – tape or hard disk – you intend to use, although this is generally less critical for their purposes than the camera type.)
In terms of pixel count, the difference between HDTV (1920x1080 pixels) and 2K (2048x1152) is relatively small. But the designs and features of most HDTV cameras derive from the television production world, and 2K cameras are designed more like film cameras. I could be wrong about this, but I believe that it is more common for 1080 cameras to record onto tape, and 2K cameras to record to hard disk. It is almost certainly possible to record either format on either kind of medium.
The 2K cameras typically shoot at 24 fps (vs. 30 fps for 1080), which will help when you make your film prints. 2K cameras are also designed to create images with a “film” look than 1080: they seem “softer” than video, and have a wider dynamic range.
In short, if your DP comes from the film world, he will almost certainly feel more comfortable shooting with a 2K camera than 1080.
OTOH, I believe that it is cheaper and easier to edit 1080 than 2K: video production houses that can handle 1080 are more common and the programs and hardware needed are somewhat less expensive. Since 2K is mostly used for cinema, it is less common and more expensive.
When it comes to exhibiting your finished film, many conventional multiplex theaters now have digital cinema projectors that are capable of showing a 2K digital cinema package (DCP). I believe that most are also capable of showing 1080 program material. But digital cinema projectors are relatively rare in non-theater auditoriums. AFAIK, no conventional 1080 projectors can show 2K.
So if you’re going to show the film in regular theaters, you could use a 2K or 1080 master (2K will look better), but if you’re going to show it non-theatrically, you’ll definitely need a 1080 master. It’s possible to shoot in one format and create a master in the other without too much difficulty, but if you’re going to exhibit primarily in one format, it makes most sense to shoot with that format.
BTW, are you committed to shooting digitally? It might not be as expensive as you think to shoot film, either 35mm or Super16. And you can still edit and master digitally if you want.