I wanted to bump this post (and change it back to unresolved) because I realize now that there are issues with how this plugin displays on mobile devices. I did not thoroughly test this out and at the time I made the comment above about it working for all sizes and settings, and I was confusing responsiveness for mobile capable.
Here are the facts.
This is a flash based player. A Flash player does not display properly, if at all, on mobile devices. The workaround that plugin developer came up with for mobile is to have the player divert to a special YouTube playlist to play the videos. This kind of defeats the purpose of having the player in the first place.
I am now testing my sites with the Opera Mobile Emulator. This obviously uses the Opera browser, but it seems to be a very good browser and I think it gives a good representation of how things display on mobile devices.
On the plugins demo page: http://www.extendyourweb.com/wordpress/ultimate-video-gallery/
I found that the player is nether responsive nor mobile capable. I tested it on Amazon Kindle Fire, Asus Eee Pad Slider, HTC Wildfire, LG Optimus 1, Motorola Droid RAZR HD, Samsung Galaxy S III, Sony Xperia Ray, and Toshbia Thrive 10.1, and on all of them, the player initially displayed okay, and linked to the YouTube playlist; but none of them would play it. They all displayed a popup window indicating that the protocol was not supported.
Interestingly, the Nokia N800, N9, and N900 Maemo all did play the videos in the special list.
In testing the player on my own website, where the player should be, it displays a rectangular banner with a title across it, and does not resemble a video player at all (no control buttons, no playlist), and nothing at all happens when it is clicked.
On a closing note, I will add that I use a plugin called FitVids for WordPress that does wonders for how a video player (any player) displays on the screen. It causes the player to fill its container size, and makes all videos completely responsive. And it makes this plugin look and perform great on a desktop display.
This is potentially a great looking plugin for desktop. It functions well and is easy to configure, and I thank the developer for bringing it to WordPress... But it does not currently work for mobile applications.