Download and install the latest version of MediaHuman YouTube Downloader. It's available for Windows, Mac OS X and Ubuntu/Linux.
Copy the link in browser. It can be a channel, playlist or video contained in the playlist. In our example it's a video from one of VEVO's playlists.
Run YouTube Downloader and paste the link by pressing the first button on the toolbar or using paste hotkey (Cmd-V on Mac, Ctrl-V on Windows and Linux). You can also drag'n'drop it from the browser.
If you added the link to the video contained in playlist, click on Add Playlist icon. And all videos from playlist will be added and ready to download.
Simply click the Start all button to begin downloading all the videos from the playlist you just added.