What is screen aspect ratio? 

The aspect ratio is the difference or ratio between the horizontal and vertical dimensions of your screen. For instance, up to now, TV sets used 4:3 screen ratios where the picture is almost square. In theatrical presentations, it has been typical to see 1:2.35 ratios, meaning that the image was presented with a width 2.35 times wider than the height of the projected image. This is the well know Panavision format. With the introduction of DVDs, we saw the 16:9 (16/9 = 1:1.78) ratio format appear, which is in-between the very wide cinematic Panavision format and the close to square TV format.

Which ratio should I choose to use? 

The aspect ratio you will choose depends on your main uses for a home theater system. If you watch a lot of TV content (including some video games still in 4:3 format) or old movies that were filmed natively in the 4:3 format, then go for a 4:3 ratio screen.

If you intend to watch manly recent movies, you may be better by choosing a 16:9 screen format and thus be ready for high definition. It is also to be noted that a wider rectangular screen such as a 16:9 or even wider screen will have a stronger impact at making the viewers immersed and feeling the cinema experience.

Also keep in mind that since there is currently many video format in use, you will probably encounter occasions at one point or another where your screen aspect ratio will not correspond to the video material you are watching. In that case, the projected image will have black bars, either over and below the image or on the left and right edges, depending on the screen format compared to the video format in use.

In the end, it may be a question of best compromise based on your main video format that you will be using. You can approach the problem from a different angle. Do you prefer a constant image width or constant height? With wide screen formats, you will get constant image height if the video format you most watch is either 16:9 or 1:2.35 (the 16:9 is probably a better compromise over all for the future as it follows HDTV resolution ratios).

The "constant width" criterion

In order to get the biggest cinematic impact, the image width is probably what will be the most important relative to the image part of a home theater presentation. Thus, our advice would be to choose your screen format to maintain a constant width, no matter what you watch.

This being said, based on the video format you most often use, various screen sizes can allow you to maintain that criterion. Let's try to summarize this:

So if your intended use includes 4:3 material, your best choice is the 4:3 screen and projector combination (the projector must use the same ratio of course).

If you never or very rarely use 4:3 material and have many 16:9 DVDs in your collection, got for a 16:9 screen and projector ratio. The 16:9 format is very attractive as it provides a good immersion feeling with a decently rectangle screen format, while at the same time offering a format that should provide long-term compatibility with the major HDTV formats (720 and 1080 resolution formats). This will allow you to use your screen for upcoming HDTV broadcast shows, next-generation video game consoles as well as many DVD movies. The only compromise here is that movies using wider formats such as 1:2.35 will have slight top / bottom black bars added. This is not much of an issue when using a high contrast screen and proper projector calibration to achieve deep black levels and good contrast. On a well tuned home theater, the inconvenient caused by these black bars will become of minor importance compared to the benefits provided by that screen ratio format.

I would not really advise a 1:2.35 screen format unless you intend to acquire a projector featuring a motorized zoom lens that you can adjust based on the material you intend to project. Keep in mind that you will have to adjust the lens zoom factor each time you change format between 1:2.35 and anything beyond that. That ratio would be only recommended for diehard home theater fans that want nothing but the best wow factor and are ready to live with the extra work it requires. 

I will add pictures to better illustrate these concepts soon...