Q: how to print from iPad to any network-attached printer (i.e. through router) without computer

iOS devices print using AirPrint. The printer must support the AirPrint protocols. See the "About AirPrint" link provided by Matt_M. It contains a list of all printers that support AirPrint.

Another option that might work is X Print Server. This is a device that connects to your network and allows iOS devices to print to network printers wirelessly. Check their website carefully because they say print to "virtually any printer." That means not all printers are supported. I have not used this device because my printer is not supported.

http://www.lantronix.com/it-management/xprintserver/xprintserver.html?utm_source =google&utm_medium=cpc&utm_campaign=xprint&utm_term=xprintserver+exact&mm_campai gn=e476bb615df94d662c50fc5571ee1c21&keyword=xprintserver+exact&utm_content=33961 809554&gclid=CI6Twcb0oMMCFRRcfgodt78A_g&gclsrc=aw.ds

Greetings Matt_M.

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printing to a non-airprinter from an iPad

To print from the iPad, your going to have to install some more software:

-- install a printer app on the ipad. many printer makers have iPad applications for their printers.  Look in the app store

-- install enabling software on a computer which you have connected the printer

Helper app -- /print n share

Print directly to the widest range of network or WiFi printers available in any app. Alternatively print via your computer to ALL printers including USB & Bluetooth printers. View read and display large PDF, Office & iWork files. Easily transfer & store files from the Cloud or your computer.

https://itunes.apple.com/us/app/print-n-share/id301656026?mt=8

Install some software on a computer that can reach the printer.  Computer needs to have wifi support

Mac software

Printopia

   http://www.ecamm.com/mac/printopia/

handyPrint  -- donationware

   http://www.netputing.com/handyprint/

Should you have some office printer that are connected to a computer there are print enablling applicaionts you can run on the iPad to print to such printers.

For Windows, try:

   O'Print

      http://oprintware.com/

   PrintCentral

     PrintCentral is the supreme printing app for the iPad, iPhone or iPod Touch

     A real DEBE -- does everything but eat.

     http://mobile.eurosmartz.com/products/printcentral.html

Couple of things, you said "Greetings Matt_M" but were responding to the OP. Was this a typo.

Second, the OP specifically said without a computer so only your first dashed item in your first paragraph would apply.