Youtube has globally changed from full-screen (4:3) to widescreen (16:9) since 2008 and this is why you cannot get them to fit on your physically 4:3 aspect ratio (square) display without them resulting in the 'black bar' on the top and the bottom of your screen.
In order to prevent this from happening anymore under virtually any and every circumstance (not just limited to Youtube) you would need a physical 16:9 (aspect ratio) or widescreen (rectangular) display in order to correctly see 16:9 aspect ratio/screen resolution settings and without any resulting 'black bars' (this is what happens when you try and use a physical 4:3 display with 16:9 screen resolution settings where the top and bottom gets cut off. Conversely, this also happens when you are using a physical 16:9 display with 4:3 screen resolution settings, there the left and right gets cut off). This is no more evident than in the case of DVD formats ('full-screen' encoded 4:3 aspect ratio and 'widescreen' encoded 16:9 aspect ratio) between the old CRT televisions and the new HDMI televisions. If you have those things, test it for yourself and you'll see exactly what I'm talking about. This is exactly what is happening on your computer sans the DVD and TV factor.
I now also feel it important to note that screen resolution settings have little to do with physical aspect ratios of displays beyond the fact that certain screen resolution settings are intended for use with one or the other physical aspect ratio. The core of your problem is not with the screen resolution settings, it is with your display's physical aspect ratio.
4:3 aspect ratio, physical or otherwise, is obsolete and it is definitely time to move to 16:9.
Now, in the same vein... Standard Definition (SD) and High Definition (HD) are yet another totally different can of worms. Your GPU claims support up to 1920 x 1080 (which is HD) but that monitor you chose can only support a screen resolution of 1600 x 900 (SD or Standard Definition). So if you also want to future-proof with HD capabilities, you'll need to chose another display that supports it (and of course, they cost more). But if that doesn't matter to you, then the ASUS VS207T-P will resolve the physical aspect ratio issue across the board.
