You can't download a printer driver app for the iPad. iOS's sandboxing won't allow an app like this to work.
However; you can solve this from the other end. You can download an app for the computer that will make any printer available to that computer show up as an AirPrint printer; another words, it will show up in the Print options for Safari and other apps that use the feature. As a bonus, you can even designate folders on the computer to print to. A quick way of exporting a PDF to the computer. If you designate a folder in your computers DropBox folder, you can easily add PDF files to DropBox by this method.
Here's some links to those apps, and a lot more info about printing. It's a bit out of date, but mostly applies.
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Some printers are directly compatible with AirPrint on the iPad. If you want this feature be absolutely certain the printer supports AirPrint before buying. Some printers may require a firmware update to work. These are usually available on the manufacturers site. If you get this kind of printer you can print directly from within apps that support the feature, which is many.
AirPrint Basics
Of course some of us already have printers and do not want to buy a new one. There are several ways to do this.
One is to use a printing app on the iPad.
Most (or all) printing apps can only print pictures and/or files that have been copied to that app via Open In or another method. Several printer manufacturers have their own apps for their printers. In general those apps will probably have the best results, quality wise.
Another method is to load a program on the computer that will emulate an AirPrint compatible printer. Your computer must be on and have the printer available to it. The cheapest (free) is Airprint Activator. More features can be had with FingerPrint (mac and windows) and Printopia (mac only).
Printopia - Ecamm Network
FingerPrint - Collobos Software
How to Install AirPrint Activator for Windows?
A few printers (mostly HP's again) can be set up with their own email address. PDF attachments can be sent via this address to be printed. The last review I saw (many months ago) said it could be quite slow (big files not recommended), and sometimes the emails got lost and never printed. The advantage was that you could literally send/print from anywhere you have an internet connection.
For small business owners and those with more than one printer who would like all their printers to work, but don't want to leave a computer on all the time. The
Lantronix XPrintServer is a small box that connects to your network and is supposed to detect and make all your wireless printers available to iOS as AirPrint printers. I haven't seen a full review on it yet, so do your research.
Some NetGear router models support NetGear Genie, which will add AirPrint support for printers on your network. If the printer is not wireless, it must be shared on the network, and the computer it is connected to must be on.