I am trying to convert mp4 and avi files to a lower resultion so they will play on my DVD player via the USB port. The USB port appears to support a maximum resolution of 580p.
What width and height is 580p resolution?
Well. your height is 580 and your width is most likely 924. This is assuming that you have a 16:9 ratio. However, your width could be 720,1280,1920,5400,900000000, etc. There's no way of telling what size the width is just based on the height. Think of Area = Width * Height. To solve this equation you will need at least two known variables. In this case, you just have one.
[Edit to properly answer your question.]
There are also differences in pixel aspect ratio (720 × 480 vs. 720 × 576) and display frame rate (29.97 vs. 25).
Above is a quote from http://en.wikipedia.org/wiki/DVD_region_code. Based on this information, it looks like you will want to most likely use 720 as your width, but will have a different aspect ratio to get to the same WxH ratio.
[Edit to add pretty shiny things.]
Math: Assuming that we are talking about a 16:9 ratio, I took a known ratio value 1280x720 (720p).
Or if you're using Windows 7, you can normally right click on the file and go to Properties. Once there click on details and you should see the Frame Width.
In this case, it is 720x400 which is about DVD quality, but a bit wider; almost 16:9
580p would be "standard" PAL/DVD signal which has a resolution of 768*576
See the wikipedia page on Computer Display Standards
As you can see it mentions it as PAL but it is also the native resolution that DVDs work at.
Just as an extra bit of information PAL is one of the common TV signal standards before HD (720 and 1080) became common, the alternative to PAL is NTSC which is used in America amongst other countries and as it only has 480 vertical scanlines has the abbreviation of 480(i/p).