Since I can remember, the aspect ratio of computer monitors has been 4:3 or 1.33 (eg, 640x480, 800x600, 1024x768). This was inherited from TV, which inherited it from (old) movies.
But movies went widescreen in the 1950s, and TV and computer monitors have been (slowly) catching up. Many newer monitors and laptops have very different aspect ratios from the above. American (NTSC) DVDs are mostly 720x480 or 3:2 (1.5). This is uncommon among displays for some reason. The one I am typing on at the moment has 1280x768 or 5:3 (1.66). Some newer machines have 1280x800, or 1440x900, or even 1920x1200 (all of which are 8:5 or 1.60).
Why all this mucking around with aspect ratios -- can't they settle on one aspect ratio, or have they still not found the perfect one? I wondered about that idly today and realised that the last three ratios I gave, while not perfect, are tending towards the perfect ratio.
Oddly enough, this "perfect ratio" (often called the "golden ratio") is not a rational number. It is the positive root of the quadratic equation x^2 - x - 1 = 0, and is roughly 1.618. Rectangles with this aspect ratio have been regarded as perfect (in particular, the most pleasing aesthetically) since ancient times. Why? Well, one reason is: take such a rectangle, and draw a line dividing it into a square and another rectangle. The new rectangle has the same aspect ratio (rotated by 90 degrees).
There are several curious facts relating to the golden ratio. I'll only mention the relevant one here (referring you to MathWorld and Wikipedia for more). If you take the Fibonacci series -- 1, 1, 2, 3, 5, 8, 13, 21, 34, ... -- and take the ratio of successive terms, it approaches the golden ratio as the terms grow larger. (This is fairly easy to prove.)
Apparently the display makers are slowly rediscovering this. 1:1 or 2:1 seem to have been passed by. NTSC DVDs are 720x480 or 3:2. For some reason this was rarely the screensize: originally they went with the TV-inherited 4:3 (which, I suppose, was two 3:2's joined along the long edge). Then they found 5:3 was better. Then they found 8:5 was better. After this they'll give us 13:8 (1950x1200 anyone?), 21:13, and so on. When they reach 89:55, they'll give us 1942x1200, which is the best you can do with an integer number of pixels at resolutions comparable to HDTV.