Q: How do I download video I've taken with my camera?  The format is .mov.

Perhaps you could try this to resolve your issue: http://support.apple.com/kb/PH12116

Importing personal photos and videos from your iOS device to your computer. - http://support.apple.com/kb/HT4083 - other useful information at https://discussions.apple.com/message/20302567

This is only a guess, since I have never used videos, but perhaps the video must be downloaded from the camera to the computer and then synced to the iPad.

I am not sure what we're answering here now.  I took it to mean the photos were taken with the iPad's camera and on its Camera Roll and need to be downloaded to a computer.  Others seem to think the OP wants to transfer from a computer to the iPad (but "downloading" means from somewhere else to the central unit, usually a computer, otherwise it is uploading).  Yet others have interpreted the question to mean transfer from a non-iPad camera directly to the iPad, in which case the answer is probably:

iPad Camera Connection Kit - http://store.apple.com/us/product/MC531ZM/A - "The iPad Camera Connection Kit gives you two ways to import photos and videos from a digital camera: using your camera’s USB cable or directly from an SD card." Can also be used to copy photos from USB drive, see https://discussions.apple.com/message/18771759

Another way. You can use a USB flash drive & the camera connection kit.

Plug the USB flash drive (works the same with an SD card) into your computer & create a new folder titled DCIM. Then put your movie/photo files into the folder. The files must have a filename with exactly 8 characters long (no spaces) plus the file extension (i.e., my-movie.mov; DSCN0164.jpg).

Now plug the flash drive into the iPad using the camera connection kit. Open the Photos app, the movie/photo files should appear & you can import. (You can not export using the camera connection kit.)

Using The iPad Camera Connection Kit

http://support.apple.com/kb/HT4101http://support.apple.com/kb/HT4101

iPad: Using iPad Camera Connector with unsupported USB devices

http://support.apple.com/kb/ht4106http://support.apple.com/kb/ht4106

Secrets of the iPad Camera Connection Kit

http://howto.cnet.com/8301-11310_39-57401068-285/secrets-of-the-ipad-camera-conn ection-kit/http://howto.cnet.com/8301-11310_39-57401068-285/secrets-of-the-ipad-camera-conn ection-kit/

 Cheers, Tom

iTunes: May be unable to transfer videos to iPhone, iPad, or iPod - http://support.apple.com/kb/TS1497 - incompatible format for devices.

You could try a WiFi transfer app. That's what I use to move photos and videos to/from iPad/computer. Photosync is the app that I use, but there are others.