Why can't iPod touch, iPad and iPhone play internet video from sites like YouTube? The reason is that these videos run on something called "Flash", which iOS devices don't support. This is the whole reason why Apple created the YouTube application on your home screen.
Luckily, some people have done their best to fix the problem.
How to play internet video on iPod, iPhone (iPhone 6S)
To play internal video on your iPad, iPhone or iPod, you can download iTransmogrify!, a bookmarklet for iPhone and iPod touch which transforms embedded Flash content into direct links to formats supported by Apple. That's to say, within a few clicks, those YouTube videos and MP3 can be played from the iPhone's native browser, say Safari.
Note: And of course there are some free video apps like YouTube for iOS you can download to view online videos and movies: Top 6 free video apps for iPad.
If you use iTransmogrify!, you are limited to several video websites actually, and the Currently supported sites include: YouTube, Blip.tv, Revver, TED talks, many embeded flash MP3 players ,Kink.fm podcasts, Flickr badges (videos are unsupported) etc.
To install this bookmarklet, just drag the link to your Safari or Firefox Bookmarks, IE users should right click and choose "Add To Favorites". After adding the link, sync your iPhone or iPod touch.
Then, whenever you find a broken video, just select the bookmark to convert it. Now, you can watch and play internet videos such as Blip.tv, Revver, TED talks and embedded TED talks, Many embeded flash MP3 players, Kink.fm podcasts, etc. on iPod touch and iPhone.