To clear things up. 'AVI' is a file format or a container. It will contain video data most likely using the codec 'Mpeg-2'. You don't really need to know this, just that AVI is the file format. IE your movies may be called movie1.avi and movie2.avi.
A DVD needs .vobs on it's disc in a folder named 'video_ts' and also menu and playing information files. Of course you are not going to have an easy time making those yourself so what you want is software that can convert mpeg-2 to DVD.
I know some software that can do this, it's called 'covertxtodvd' where X represents a large number of different file formats and codecs that it can convert to a DVD.
This software can also prepare all the menus and playing information files and even burn the resultant files to DVD and then trash the generated files, keeping your originals (because when you encode you always lose quality so originals are always the best).
so look for a convert to DVD type software or more specifically 'convertxtodvd'.
Heres a link: http://www.vso-software.fr/products/convert_x_to_dvd/
Added: After re-reading your question, I am confused. Do you mean DVD-Player or DVD-Reader. Because a player is designed just for Video DVD's and outputs a signal to a Television whereas a reader is designed for ANY files to be read from the DVD and plugs into a computer.
In the case you want one for a Television that can PLAY the DVD's then some DVD's can read .avi if the codec is 'divx' or 'xvid' or some even 'h264'. For converting your files to any of these formats you should use the software 'Super' by 'erightsoft' (http://www.erightsoft.com/SUPER.html) and maybe google for a tutorial if you need to know how to use it to encode in Divx.
This IS a tricky business though as depending on the release date of the specific codec, it may have different data put into the file, which shows electronic devices what it is. IE: Your DVD player even if it reads Divx, may not read the version you have converted your files to.