YouTube is pretty strict about its stance on not allowing downloads of its videos. Even many popular browser plugins that once supported downloading of YouTube videos are now blocked from the site, so it should come as no surprise that any apps that once allowed you to do the same have been patched to remove the ability. That being said, there's still a way to do it, and here's how.
First, you'll need to download a free app called Titan Downloader. This was one of the apps that once allowed super easy downloading of YouTube videos, but has since blocked the feature -- on the surface, at least.
If you browse for a YouTube video directly from the app and attempt to download, you'll be met with a warning that, due to "copyright issues," the app doesn't allow downloading on YouTube.
Instead, use Safari to find the video you wish to download and copy the url.
Now, back in Titan Downloader, paste the address and bring up the video. Click the play button so the video begins to play, and then quickly tap the "Done" button in the upper left corner.
As soon as the video window shrinks down, tap and hold your finger on the video window itself. After a moment, a menu will appear. Tap either "Download" or "Download As" and the video will be saved within the app.
Now, if you wish to save the video directly to your phone, tap the "Videos" tab on the bottom of the screen, tap and hold the video icon and then click "Save to Camera Roll." Tada!
I wouldn't be surprised if this hidden (or overlooked) bit of functionality was patched out before too long, so if downloading YouTube videos is something you want to be able to do long-term, and without jailbreaking your device, make sure you download the app with a quickness.