You can use an adapter and a cable to mirror exactly what's on your iPad's display onto a projector or television screen. When picking out an adapter, you need to know if your device has a Lightning or 30-pin Dock connector. New iPads and iPad minis have a Lightning connector. iPad 2 has a Dock connector.
You also need to know if your projector has HDMI or VGA input. Use this flowchart to help you choose from among the four possible adapters.
Identify the type of connector on the bottom of the device and type of cable used by the projector or television.
You can find most of these adapters on Amazon.com, Apple.com, and other sites. You might find cheap versions made by a manufacturer other than Apple. I have haven't tried a non-Apple iPad adapter. From the reviews I've read, I see that some of them do not work properly. To get a discounted adapter, try finding a used Apple adapter on Amazon or eBay.
Sometimes people complain about the length of the adapter. An adapter is not a cable, so it connects to the HDMI or VGA cable that comes from the projector or TV. If your existing HDMI or VGA cable isn't long enough, you can get a longer one for cheap from Amazon.
If you'd rather not be tethered by a cable, there are wireless solutions, including Apple TV, AirServer, and Reflector. See my post 10 Ways to Show Your iPad on a Projector Screen for a comparison of wired and wireless mirroring options.