Patrick,
While you can upload an MXD to ArcGIS.com, you need to remember that the MXD doesn't store any data, but rather references data. So, if the person you want to download your MXD has access to the same data (e.g., they work with you and have access to local directories), they will be able to access the MXD from ArcGIS.com, download it and open it.
However, if the person doesn't have access to your local data, then you need to include the data with your upload. At 9.3.1, you can upload a layer with it's associated data, creating what's called a layer package. To do this, right-click the layer in ArcMap's TOC and click Create layer package. Remember that the data will get packaged up inside the layer package, so if it's a large data source, the layer package (.lpk) file will be large too.
Once you upload your layer package, you'll need to share it within ArcGIS.com so others can access it. You can share it with everyone (make it public) or share it with other members in a group. (Note: make sure you change the setting on the website at the top of the page to "Show:All Content."
ArcGIS 10 has the ability to create Map Packages that package up an entire map.
One last point... Layer packages are not viewable on the web, for example, display over the ArcGIS.com map. For this, you need to reference a map service.
To learn more, search the ArcGIS Desktop help for "layer package"
Hope this helps,
Mike
