A majority of high end wide-screen monitors (16:9) that are titled as "Full HD" have a native resolution of 1920 x 1080 (i.e. 1080p)
But there are a select few "Full HD" monitors that have a slightly higher resolution.
i.e. this Dell 23" boasting a 2048 x 1152 resolution.
So is this higher resolution better?
Will a monitor with a higher native resolution work well being set at 1920 x 1080?
LCD monitors look best when set to their "native resolution."
If the Dell LCD monitor is 2048 x 1152, then the video resolution should also be set to 2048 x 1152. If you set it to 1920 x 1080, it will not look as good.
It really depends what you are trying to do with the monitor. If you are using it to run applications, the higher resolution will help a bit. You'll be able to see slightly more. Realistically though, the advantage is pretty minimal.
If you are using it to watch HD video, the 1920x1080 monitor will have a slight edge. To make 1920x1080 video fit onto a 2048x1152 screen, it will need to be resized. This will cause slight blurring (probably not so you notice) and more importantly, it will take a lot of CPU/GPU to do the resizing. That's processing power that can't be used for other things.
As to the second question, you want to run the LCD at its native resolution. It will always display 2048x1152 pixels so feeding it a smaller picture means that picture needs to be stretched out to fit. The scaler in typical monitors isn't all that great and it will likely become fuzzy.
Since on an LCD monitor, each pixel turns out having a dedicated element that makes it glow, thus the true (or native) resolution of any LCD monitor will be the same as the number of such dedicated elements.
If you try to assign a resolution other than the native resolution, then it will have to mix some of the pixels around and merge parts of them into adjacent elements, thereby giving an overall fuzzy look to the image on the screen.
Having a larger resolution means that you can have more space on your desktop, for example you can comfortably read two pages in word side-by-side. Besides having more workspace area, having a larger resolution does not really have any other specific advantage.
Always try to run a monitor at its highest resolution (that's also the recommended resolution). Having a high resolution on a small screen (compared to a TV) gives you a smoother image (it's considered that at one point the image looks as smooth as on paper).
However. if you feel the text and the icons are too small at large resolutions, consider increasing the DPI from Windows or Mac OS. This way, the text and GUI are larger, but they all look a lot crisper.
LCD monitor should always run at its highest resolution. So, watching 1080p video on a 2048 x 1152 monitor will DECREASE the quality. Same for the fullscreen games (unless you run them at 2048 x 1152 in which case they'll look great, provided you have enough horsepower).