How do I access an external drive mounted on a machine on my own network?

Here is an easy GUI solution:

On the server, where we have attached the USB drive open Nautilus and browse to the mount point of the USB drive (usually found in /media/). In the right click context menu on this folder open Sharing Options, tick Share this folder, give a sensible name for the share, and Create Share

On the remote, i.e. your laptop, open Nautilus to browse the Network for the share as named above. On mouse double-click this will be mounted as a network drive on your laptop, and it will appear as an icon on the desktop.

To mount a samba share as non root user in your home directory to have access from all applications you may want to have a look at smbnetfs.

In case you did not set your USB-drive to auto-mount you will be able to mount it remotely by using SSH (after having installed openssh-server on your desktop).

You can use SFTP, which uses the more secure ssh (encrypts everything, including login credentials), and seems to be faster than the protocols that Takkat's answer uses.

First, on the machine with the Disk connected to it (the Desktop in this case), install the ssh package.

Then on the machine you want to access the files from (the laptop in this case), Open nautilus. Click the File menu and then Connect to server... For service type, select SSH and put in the IP address of the computer the disk is connected to for the Server field. Optionally, you can check Add bookmark and enter a name to have it always show up in the sidebar. Click connect. You will be asked for your username and password for the Desktop. You will be able to see all the files on your Desktop and access your drive remotely under the /media/ folder.

The remote filesystem is mounted in /home/yourusername/.gvfs/, so you can can use command line tools too. For example, on the laptop: nano /home/avilella/.gvfs/sftp_for_desktop/media/externaldrive/some_text_file.

If the drive isn't already mounted, you can have the Desktop mount it so you can use it by running ssh [email protected] from the laptop (assuming avilella is your desktop username and 192.168.0.2 is the desktop IP). It will ask for your password and then you will have a command prompt for the desktop. Then you can run sudo fdisk -l or sudo blkid to identify which /dev/sdXX partition is your external drive. Then you can run

sudo mkdir -p /media/externaldrive
sudo mount /dev/sdXX /media/externaldrive