Which monitor should I buy?

To get 1920x1200 from the Hannspree, I think you'd need to use DVI rather than HDMI, and it might not have an option of whether to add black bars to a 16:9 signal such as from the PS3 (as opposed to stretching it to 16:10)

As far as other monitors, I use a P2770HD and I'm thinking of getting a Sceptre E320BV-FHDD in a few months, so you might consider those. However, if you can provide more information like what size monitor you're used to (or what size range you want to consider) and your budget, I might be able to provide better suggestions. You could even fill in a copy of this template that I put together last year:
http://www.tomshardware.com/forum/59847-3-template-moni...

Ya thanks, I will defiantly check out that template. So is DVI better than HDMI? Because I always assumed HDMI gave the utmost best signal possible. Also, about the PS3, are you saying that it will stretch the image to fit the screen rather than having the black bars?

Also another thing, I will be sitting pretty much right in front of it to play games and watch movies. Is 23 to 24 inches too small, too big, or a good size? Also, will they display images better than say an HDTV would?

There's no difference in quality between DVI and HDMI, but for resolutions above 1920x1080, you usually have to use dual-link DVI cables (more bandwidth - up to 2560x1600 is possible). As far as the PS3, a good monitor will give you the choice of whether to stretch the image or use black bars, but Hannspree doesn't provide an online manual that I can find for the model you're looking at, so for all I know it could lock you into one or the other. 1920x1080 is already the correct aspect ratio for the PS3 signal, so you don't need such a choice.

The size is very subjective - some people seem content to stick with 19 inch monitors, but others (like me) prefer larger monitors - 26 to 32 inches is my desired range (you might have noticed that the monitor I mentioned as having now is 27 inches, and the TV I'm planning to get later is 32 inches).

I'm not sure there's really that much difference in quality between a monitor and a TV anymore, at least if the TV is full HD (1920x1080). Some sizes and features are easier to find in a screen sold as a monitor, and some are easier to find in a screen sold as a TV, but there's a fair amount of overlap.

So what Is your opinion about these two monitors? Are they good for what I am looking for?

http://www.newegg.com/Product/Product.aspx?Item=N82E168...
Or
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Also another question: can you explain about DVI. I mean I know that it is a cable to transmit video, but how does it compare to HDMI? And is it analog or digital? Any thing else about DVI would also be helpful, thanks.

DVI is capable of carrying both analog and digital, though HDMI also carries audio. I think Wikipedia could explain them better than I could, though.

Both of those monitors would probably work for you, but the VG236H is considerably more expensive, so unless you're sure you want active 3d with shutter glasses (which will require dual-link DVI to pull off at 1920x1080 - HDMI only supports 120hz for 1280x720), I suggest you skip that model.

But won't 250 cd/m2 be too dark to watch blueray movies and other 1080p content? And about DVI, if it only transmits video, then how do I transmit the audio too?

The standard target when calibrating a monitor is 120 cd/m2, which is usually enough in a well-lit room. If you prefer to turn out the lights to watch a movie, you might not even need that much brightness. With DVI, you transmit the audio through separate cables (and often to separate speakers anyway).

So in that case would it be a good decision to get Asus VH238H Black 23" Full HD HDMI as my monitor considering what I will be doing? (i.e. playing the latest games, watching movies in 1080p, and other Blue Ray movies etc.) Or would you suggest a better monitor around 23" -24"?

I notice that the VH238H is only rated at 4 of 5 eggs at Newegg. The best rated monitor at Newegg based on the requirements you've given so far is the Asus VH242H:
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
If you really want an LED backlight, the best rated at Newegg is the Asus VS248H-P:
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Well I mean LED backlight is pretty much necessary, right?

The LED backlight has its advantages, such as allowing the monitor to be slimmer and usually have lower power consumption, but it isn't a requirement for anything else that I'm aware of.