How to view movies stored on external drive on my iPad?
You can connect USB devices to your iPad with one of Apple's USB camera connection adapters, but there are a number of limitations and requirements to make this work.

There are also drives now that are specifically designed to work with Lightning connectors — if you are viewing this page from a Google search and don't have a drive yet and want to see if you can make this work, you might want to look into buying one of them instead.

If you already have a non-Lightning USB drive and want to use it with your iPad, read on.

Limitations


The drive must either be externally powered (plugs into the wall) or require only a tiny amount of power. I wasn't able to find exactly how much power the port is allowed to supply, but it almost certainly won't be enough for a hard drive — flash storage/solid state only.


Note: It might be possible to skirt this requirement by using a USB hub which draws power from an external power supply. I wasn't able to find definitive information about whether this works, though, so do some testing before buying anything.

The drive must be formatted in a certain way. I don't have the kit myself, so I can't test this, but some sources claim it must be FAT/FAT32/exFAT formatted and others that it must be HFS+ formatted. NTFS definitely won't work. (This may be a variable dependent on your iOS version — test it out with another drive first that you don't mind reformatting a couple times.)


Establishing the physical connection

If you have an older iPad with the 30-pin dock connector (1st, 2nd, or 3rd generation iPad), you will need the iPad Camera Connection Kit (discontinued, but available from Amazon or other places).

If you have a newer iPad with a Lightning connector (any iPad Mini, Pro, Air, or the 4th generation iPad), you will need the Lightning to USB Camera Adapter ($29) or the Lightning to USB 3 Camera Adapter ($39). The more expensive one's only advantages are that it will allow you to charge your iPad simultaneously, and if you have a 12.9" iPad Pro it will allow you to transfer data with USB 3 speeds.

If your iPad is jailbroken

If you're jailbroken, it's pretty easy. Simply use a file manager like Filza or iFile and view the files from there, or a media viewer like Kodi (formerly XBMC).

If your iPad is not jailbroken

If you aren't jailbroken, it is still possible to view movies on an external drive, though it necessitates that you have enough spare storage on your iPad to hold a movie, that it's fine to either reorganize or duplicate your movies, and that all your movies are already in iOS-readable formats (I wasn't able to find a list of supported formats for this, but it's a fair bet that MKVs are out, for instance).


On your computer, copy the movies you want to watch to a "DCIM" folder at the root of the drive.
You may need to also rename the movies to something like "DCIM_1234" — sources vary as to whether this is a required step.
Connect the drive to your iPad with the adapter and open the Photos app.
It should now prompt you to import your media.
Select the movie(s) you want to copy to your iPad and tell them to import.
Once the import has finished, you can unplug the drive from your iPad (no "eject" button is necessary).
Your movies will be in the Photos app as though they were videos you took with a camera. You will need to watch them in Photos or in a third party app which can view media in your Photos library.


If any of the requirements of that method aren't an option, there may still be a way to do it with a third-party app, though I wasn't able to find any options in my research while writing this answer. If I find any further information about this in the future, I will update my answer to include it.