The image that a projector can produce is created by one or three chips inside the projector. Commonly these are either LCD or DLP chips.
These chips are responsible for the resolution of the projected image, or more specifically, for the resolution that the projector can send out through its optical system.
After that other factors determine what the audience sees and how they perceive it. These will need to be looked at separately, but examples of these other factors would be the projection material, the viewing angle, the ambient lighting levels, and the size and ratio of the surface.

It is important to understand that the resolution of any projector is fixed. It is whatever the resolution of the chip or chips inside the projector is. Terms like “HD compatible” or the list of "compatible resolutions" do not change the inherently fixed resolution of the projector. It is equally important to understand that there are very few display ratios that these chips are available in. More information about display ratios can be found here.

In the specifications or technical facts about any given projector, this fixed resolution is commonly referred to as the native resolution. Here is an example of what this means in practice:

Let’s say despite the fact that I have a good idea of what I need for the task at hand, the fictional theater I work for asks me to use their 3-chip LCD projector with native XGA resolution and a brightness rating of 10,000 Ansi lumens. Projectors like these are quite common because they are small, bright and relatively cheap. As mentioned above I already know that I would like to use a 16:9 DLP projector with about 7,000 ANSI lumens, but the theater wants me to use their projector because they have it and it won’t cost any extra money. Hence I need to work out how this projector will affect the imagery on the surface that has been provided.

My surface is 24’ x 13.5’, which exactly matches a display ratio of 16:9. The projector I have been offered has a native resolution and chip that is 1024x768 (XGA) which is a 4:3 ratio. How do I best use that ratio on my 16:9 surface and what will the consequence be?

I will inevitably over-project, because my surface does not neatly fit into the ratio of the projector.

As a consequence I waste vertical resolution, because I have to fill the width of my surface. How many pixels do I waste? Well, that would be determined by the horizontal resolution of the chip divided by the ratio of the projection surface. The projection surface ratio is 16:9 - in other words sixteen divided by nine which is 1.77778. So the horizontal pixels that end up on my 16:9 surface if I fill the width with 1024 pixels are 1024 divided by 1.77778, which is 576 pixels (rounded up).
Consequently the resolution I can use on my surface if I choose to go with the venue’s XGA projector is only 1024x576. I throw away 192 pixels of vertical resolution or 192 x 1024 pixels total.

At this point somebody who works for the fictitious theater points out that the projector is HD compatible and lists 1920x1080 as a compatible resolution. So in his or her mind resolution should not be a problem. However, there is a lot of confusion about what people understand "HD compatible" to mean in this context.

In my experience, this is what happens: "HD compatible" means that I can send a HD signal to that projector and it will use it. By "use it" I mean the HD signal will be processed inside the projector to display on the internal 4:3 XGA chip. In other words it will be scaled to fit. There are usually two ways this scaling is done: cropped to fit or letter-boxed, which are solutions to fit either the width (horizontal pixels) or the height (vertical pixels) of the incoming signal onto the chip.

Either way the resolution will be limited by what the 1024x768 chip can display, and a large amount of the orginal HD resolution will be thrown away.

Now let's work out the brightness I would end up with should I use this projector on my surface:
We are fitting 4:3 square area over a 16:9 square area by filling the width of the 16:9 area.

Whatever part of the 4:3 area’s height does not fit into the 16:9 area’s height will be wasted.
We have to work out how much, e.g. what percentage, of 4:3 does the 16:9 area make up if we stretch the 16:9 area to the 4:3 width.

Mathematically speaking the question is what is the factor x that makes 3/4 = 9/16. The answer is 3/4 or 0.75. Expressed in percentage that’s 75%. Hence only 75% of the brightness and resolution of the proposed projector are used. 75% of 10,000 Ansi lumens is still within the target of 7,000, but the low resolution is a problem for the photographic imagery I have been asked to show on the screen.

There are multiple ways to resolve this equation, but for the purpose of this little article, I am not going to go into more mathematical detail. It is more important to note that this can be worked out without too much head scratching, and the result may be important to know for the design to succeed.

The way to answer these questions involves more math than many people I have talked to or worked with are comfortable using. Nevertheless these are necessary calculations and values if you want to make an educated choice about which projector will work for a given project or how a given projector will affect your design. So somebody has to work this out - the limitations of the technology and the science involved do not change. It is also worthwhile to remember that the projection designer is responsible for the success of and look of the projections, so it is a good idea to work out whether the equipment will degrade the design or support it.

In this example I would decline to use the 10,000 lumens XGA projector offered by the theater with the argument that the resolution will not be sufficient for the intended imagery, and the director and designer will find the resulting image quality unacceptable.

For reference, the iphone 4 takes pictures at a resolution of 2592 x 1936 and lower end digital cinema cameras have a resolution of 2048×1080 (2K). Those are the kind of resolutions consumers and theater goers are used to seing. Therefore a visible resolution of 1024x576 will read as low compared to what we are aused to seing.

I prefer to use a projector with a 16:9 chip, a resolution of 1920x1080, and about 7,000 ANSI lumens. This way I can use the entire resolution on my surface and waste no brightness by over-projecting. The 1920x1080 resultion will allow me to show sufficient detail on the surface for the photographic imagery to read well.

I believe that a low resolution look should be an artistic choice or a choice made for budgetary reasons and not a result of poor or uneducated euipment choices. That would not be design, but an accident.