How can I print out a document from my iPad to an AirPort- connected printer?
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You have a few third-party options, but all of them require that your Mac is running in order to print.

The one we use is Ecamm Network’s $19.95 (£13) Printopia 2 (www.ecamm.com), a printing utility that works with all iOS devices. First, you’ll need to make sure that your Mac can see the printer on the network. If the printer doesn’t appear in the Print & Fax (Snow Leopard) or Print & Scan (Lion) preference pane, click the + (plus sign) at the bottom of that preference pane and add it. It should appear as a Bonjour device. Printopia appears as an item within System Preferences. Select it and, in its printer list, enable those printers connected to your Mac – your AirPort-attached printer should appear in this list. You can now print to it from your iOS device. Printopia also allows you to send files directly to your Mac as well as place them in your Dropbox folder.

Other options include Netputing’s AirPrint Activator, (donation requested, www.netputing.com). This requires that you share your printer using Mac OS X’s Printer Sharing feature. There’s also Collobos Software’s $10 (£6) FingerPrint (www.collobos.com), which offers many of the same features as Printopia and is also available for Windows PCs, though it supports fewer printers.