ms2007+ uses a different 'format' for the files than what was previously used. So if you try to open a 2007 doc on a pc that does not have the 2007 suite or the previously mentioned add-in installed, the pc will not recognize the 07 doc file at all. it will see that theres a file, but it will not know what program it belongs to.
as already mentioned, you can A) safe the file as a 03 format. this will open just fine on your 03 pc. but you will also have multpile versions for each file. or B) install the add-on on the 03 pc and keep all your files in the 07 format. NOTE: when you open a 03 formatted file in 07, you'll see 'compatibility mode' in the title bar. completely normal... doesnt really mean anything other than some options that were not available in 03 will not be available; unelss you convert it to a 07 formatted file.
file sizes of the 07 format are smaller than when saved in the 03 format...for whatever thats worth to you.
another thing to think about... is if you go with option B (the plug-in) and send this file to anyone else who may also be using MS03...they will not be able to view the document unless they too install the plug-in).