How do I use a printer hooked up to an Airport Express unit?

Your timing is excellent: I just recently set up one of these networks myself and found it rather tricky, truth be told. What I ended up doing was reseting the Airport Express three or four times (which was a pain! Fortunately, it's documented here) and fiddling with various settings until I got everything to work properly.

The first step you'll need to do is make sure that your printer and Airport Express are being plugged in reasonably close to your Ethernet network hub or router. While it should theoretically be possible to have the device just serve up your printer on its own wireless network, I found that the Airport Express was cranky about starting up on a power cycle without a live Ethernet plug. Then plug in your printer via the USB connection, then cycle power on your printer after you've plugged it into the Airport Express.

Hopefully your Airport Express will display the cheery green light meaning all is well, and have identified the type of printer you've hooked up. You can check that by connecting to the Express network (which might be called something like "Apple Network 0c0037" or similar) and launching Airport Admin Utility. Start that and it should immediately see your device:

Mine has the snazzy name "Base Station 0064c3", as you can see. I select it, enter the password (I strongly recommend you set an admin password so no-one else can monkey with your unit, btw!) and now I see this:

Worth noting is that this is where I can change the administrative password (click on "Change password..."), change the broadcast name of the network (mine's called "hardcopy" since it's for the printer device) and add a password to the wireless network you're creating (click "Change Wireless Security...").

Make all the changes you desire, then click on "Update" and let the device restart with its new configuration options. Probably, you'll need to reselect it the first time if you've renamed the network, but that's easily accomplished. Once you're online, go back into the Airport Admin Utility and select the Airport Express again. This time, choose View --> Summary and see if your printer is recognized:

If your printer isn't recognized, check out the help material from Apple to remedy the problem, but most likely it'll be fine.

Almost done, believe it or not. Now you can quit the Airport Admin Utility because you want to launch a different program, Printer Setup Utility. Start it, click "Add+" to add a new printer, and select "Bonjour" if it isn't automatically selected. Your printer magically shows up:

There's my printer, the Samsung ML-1740. I simply select it, click "Add", and it's now a new printer I can select when printing from any Mac application. Not too difficult once you know the steps!