Ah, you're getting mixed up between two Adobe products:
Adobe Reader is a program for reading .pdf files.
Adobe Flash is a plugin which allows you to watch flash videos and play flash games - this is the one you'll want.
This link will take you to the download page for it: http://get.adobe.com/flashplayer/
It should detect what internet browser and operating system you're using automatically.
After installing that, you should be able to watch YouTube videos without any trouble.