Downloading youtube videos is possible through one of the many tools and websites, but directly from the browser like in Safari browser, or firefox browser there is some shortcut needed which can make the download of youtube videos happen in the quicker and easier way.

Some of the sites needs you to copy the youtube video URL, paste in the download client site and then start the download. Instead, for Google Chrome, there is a bookmarklet which can help you download youtube videos in easier fashion.
Its so-called as YouTube Videos Download plugin, but its not any 3rd party plugin as the plugins/extensions as in firefox are not available in Google chrome browser.

So, to download the youtube videos while watching them, install this plugin to get the downloads easily.

For firefox browser, the abundance in availability of add-ons for these activities makes it easy, but till Chrome gets the add-ons feature in future, the best option to download through the browser is this plugin.

Here is how you can download youtube videos on safari browser and list of tools that can help you download youtube videos.