First off, notify-send is a nifty utility you can use to make informative boxes pop up on your screen. See the Ubuntu Manpage for more information on that. Now what I wanted is to trigger those notifications remotely. After some checking around, I found this is rather simple.

First off, log in on the computer with

ssh -X user@remotehost

Once logged in, set the display and then issue the commands

notify-send "Hello World" "I made it!"

Things to note: this assumes :0 is the display being used as well as that the user you log in with has the correct permissions to issue the command.