Welcome to the forums Jelly Bean. First off, it is an S4, not a 4S ;-) don't wanna confuse this with some fruity product
Second, it depends on the type of video. So far I've found that when I play some videos it pops up 'video player' or 'downloads' and I choose the action at that point. 'Downloads' obviously d/l the video straight to your phone (or SD card) and 'video player' just streams it.
As Rushmore mentioned above, an app is the best way to d/l a video that doesn't prompt those two options I mentioned previously. Two options: Viz (aka Video Downloader) and Free Video Downloader (FVD). Both have over 1,000,000 downloads and at least a 4 star rating.
Hope this helps
edit: upon further review, I don't think FVD supports Flash. And neither App can d/l from YouTube. I'm assuming NONE of the apps available will be able to d/l YouTube though..