Starting with Honeycomb the storage partition is exposed through MTP (or PTP), this way it can be accessed concurrently by both Android and your computer. Before Honeycomb, said partition could be accessed as an USB mass-storage drive but that required it to be unmounted, thus making all data unavailable to Android while it was connected to a computer.

In my experience only Windows 7 has passable MTP support. On Linux I had mixed results with gvfs and mtpfs (slow and crashy). jmtpfs [1] is a nice replacement, though. Google also released an application for OS X [2] but I can't try it since I don't own an Apple computer.

My solution to the problem was to install a WebDAV server (such as [3]) on my device (Galaxy Nexus) and access it wirelessly.

  [1]: http://research.jacquette.com/jmtpfs-exchanging-files-between-android-devices-and-linux/
  [2]: http://www.android.com/filetransfer/
  [3]: https://play.google.com/store/apps/details?id=com.theolivetree.webdavserver