I have a HP Officejet J3680 Printer with a USB connection. How do I install it on Ubuntu server? Thank you.

The easiest way to do it is to use HPLIP, which I believe is included in Ubuntu desktop; I'm not sure about Ubuntu server.

If HPLIP's not already installed, it's in the repos.

this seems to indicate you're out of luck (hover over the "I don't see my operating system").
I recently installed a Deskjet 3050 J610a and it had a (good) Linux driver on offer. More up-to-date (scanning particularly) than the Ubuntu offering.

I know it installs on Ubuntu because it automatically installed on a Ubuntu 10 desktop, without any issues. Any ideas would be appreciated.

Here's the listing on the Open Printing Database site. It indicates that the printer works under Linux.

Note that HPLIP is not a driver. It is an HP utility for managing printers; using it has the advantage that it will select the optimal driver for that printer. You can manage HP printers without it, but it makes like easier.

The title of your posts mentions using Ubuntu 11.04. The HPLIP may not have caught up to that version of Ubuntu yet, which may be part of the confusion factor. When you download HPLIP, it will extract as a set of files and an executable script. When you execute the script, it will ask several questions and then compile per your system. I agree with frankbell that this is the best way to proceed.

The title of your post also says that you are using Ubuntu SERVER. If you mean that you are using the server edition, with no gui, you may have to adjust the configuration script. As frankbell also pointed out, it is an hp tool set. One of the components will be to install gui based pop ups and tray icons that a server edition won't have. When you run the hplip script, it may detect this automatically and it may not, but it is something to keep in mind as you try it.

http://hplipopensource.com/hplip-web...00_series.html
Here it's written that your HP printer is supported partially.

Look at the version of hplip in Ubuntu 11.04. If the version on the website is newer, remove yours and install that one, the latest, from the website.
The howto is also available there.