In addition to Russell's excellent answer, I want to point out that there isn't just a single resolution to think about.
The projector will have a single native resolution, like 1024 x 768 on old or cheap models. However, the resolution setting of the computer driving the projector also matters a lot. Many projectors will auto-scale, basically resampling a video feed of different resolution into their native resolution. It's not good enough just to size your picture to maximally fit within a 1024 x 768 (or whatever your projector uses) landscape area, but the computer must also be set up to produce video at exactly that setting, and you have to make sure your picture is displayed on the full computer screen.
For example, let's say the projector is 1024 x 768, and your original picture is 4266 x 2844. Noting your picture aspect ratio is wider than the projector's, you scale the width to fit and the height comes out to what it comes out to. In this case, you make a version of your picture that is 1024 x 683 pixels in size.
So far so good, but that's just one step. Next you have to make sure the computer actually drives each of the projector's pixels directly without something doing resampling. That means you don't just connect the computer to the projector and randomly fiddle until you see something come out on the screen. You have to make sure the computer display settings exactly match the projector. This means deliberately setting it to 1024 x 768. Maybe you still get a picture with the computer set to 1280 x 1024, but it won't look good since the projector is resampling it to the 1024 x 768 it ultimately needs internally.
Two steps down, but you're still not done. You have to make sure your carefully sized 1024 x 683 picture is actually displayed across the entire computer screen, else the computer will be resampling your picture before sending it to the projector and it still won't look good. This means you don't just load the picture from a web browser or something. Those will put a border around the display window, and therefore show your picture in less than the 1024 x 683 pixels it is intended to be shown on. You have to use some software that is capable of showing your picture and only your picture on the screen, preferably with a black background. Such software certainly exists (including my own IMAGE_DISP program), but it's probably not your web browser.
So to recap:
- Make a version of your picture that maximally fits within the projector's native resolution.
- Make sure the computer output matches the projector's native resolution.
- Make sure your picture is displayed on the full screen of the computer.
Skipping any one of these steps will significantly degrade the final appearance.