Is it possible to download YouTube Videos on my New Samsung Galaxy Y android phone and have it stored in my SD card?
Yes. You can take either of the following two approaches:
Use tubemate. With this app you can browse YouTube and download the video just like you would download from websites like keepvid.com.
The latest version of YouTube allows the videos that you opened or YouTube thinks you'll watch or have added to watch later queue to be downloaded in background so that you can watch it later. Read here. -> The preload option is no longer available in the official youtube app since version 5.6, see google support note here
For the first time, YouTube will start downloading videos it thinks you might want to watch, whenever you’re plugged in and on Wi-fi.
That’s great news for commuters, for example, who can now watch videos when outside their mobile coverage zone (or without paying for costly video downloads over 3G).
YouTube, careful to draw a distinction between this and total downloading of its vids (which breaks its terms of service), is careful to point out that you’ll still need to be online for the first second when you press play.
I used to use Tubemate. It has many options, one of which is the ability to download the audio part alone so that you can also just listen to the video rather than watch it.
I later implemented a YouTube downloader script in my browser app. It has a flaw though - new videos loaded in the same page will not update the download link.
The Google Play version of the app has not been updated for a long while and may not work. However, with the desktop browser disguise option and a Greasemonkey script, it might be possible to make it work in a mobile device as well.