Use a target attribute in your anchor.
target="_blank" should do the trick.
<a target="_blank" href="daily-cat-video">click to see the cats</a>
If you want to get fancier and include window sizes and things, look at
Opening new window in HTML for target="_blank"
an window.open.
. this will let you have more control over the window that gets open.
If you are in your mobile app, then opening in the youtube app is normal for android. It looks at the intent and will present the user with the options that can handle that intent.
If it gets a youtube video, it will ask the user if they want to open it in youtube 'just once, or always' as well as give the user other apps that can handle the intent too, i.e chrome,