Everyone has done a good job of explaining but since you're asking basic questions I wouldn't be suprised if you're a bit confused by the technical answers people are giving you. I'm going to do my best to reword the answers and add my understanding, hopefully it'll help.
1. What exactly does resolution do?
To make up an image on a screen, heaps of tiny little dots are activated in a pattern making up the words and pictures, everything that you see on the screen.
As other people have said, "resolution" controls the exact amount of dots that are available for your video card to use to produce the image.
By increasing the resolution, you increase the total number of dots on the screen and decrease the size of each individual dot.
This as you may have noticed has two side effects: everything on the screen gets a bit smaller, but, you can fit more text, images, icons etc onto the screen.
By increasing the resolution everything will probably look a bit clearer since the dots (pixels) are a lot smaller. Try using 640×480 (if it'll let you) or 800×600 to see what I mean about having big pixels (look at text and notice how it looks blocky and curves are not smooth but have a staircase effect). Increasing the resolution gets rid of these problems, plus you get the convenience of extra screen space since everything is smaller and you won't end up with a taskbar that takes up half the screen or a dialog box that takes up three quaters of the screen.
2. Why dont games support the resolution I run at?
While it is generally much more useful to run your computer at a higher resolution it does demand a lot more work on your video card.
What you'll find is a lot of older games such as Half-Life were developed at a time when video cards were not capable of running at a very high resolution. Consequencely the game developers did not bother writing support for such higher resolutions because the technology (of the time) did not support it. Almost all of todays newer games are all written with support for these higher resolutions such as 1600 or even higher in games like Call of Duty because high-end video cards such as yours are capable of running these high resolutions smoothly.
3. How come the higher the resolution the less FPS i get in games like UT 2004, I average around 40 FPS?
I think this has been explained quite well, but I'll just briefly reilliterate.
When you increase the resolution, you increase the number of dots that the video card has to produce to make a single frame. Because of this, it takes a lot more processing power to try and produce a single frame at a higher resolution than it does to produce a single frame at a lower resolution.
The side effect is that less frames can be produced at a higher resolution each second than at a lower resolution, therefore the fps (frames per second) is lower in higher resolutions.
If you have a good video card then even putting the resolution up to the maximum supported resolution, your card will still be powerful enough to produce enough frames every second to keep the game looking smooth and not like a disjointed picture show. (Generally about 40fps or greater looks smooth)
4. What is the monitor refresh rate and how does it affect my game? I currently have a 60 hz refresh for my resolution.
As has been stated, "monitor refresh rate" refers to the amount of times your monitor updates itself every second. You could almost think of it the same way you would the frame rate in a game but in this case the focus it not so much what your video card can handle but what your monitor can handle.
If you use a slow refresh rate like 60hz then your monitor will basically be flickering like crazy instead of producing a beautiful clean image. You may not notice it but if you stare at a monitor for many hours a day and get used to a higher refresh rate you can really tell when you try and go back to a lower one. I mean I can tell the difference between a monitor running at 60Hz and a monitor running at 75Hz simply by looking at it. It actually becomes quite annoying to stare at a monitor at 60Hz after you're used to a higher refresh rate.
If you wanna check this out for yourself put up some sort of a white screen on your monitor, a blank notepad window for instance, and stare slightly to the left or right of your monitor, at the wall behind it if there is one. Now looking out of the corner of your eye observe your monitor and try and see if it is "flickering".
Unfortunately I suspect that by running your monitor at such a high resolution it will be incapable of producing a refresh rate higher than 60Hz. You could always try to increase it though. (Change refresh rate by: Right click desktop > Properties > Settings > Advanced > "Monitor" Tab* - this may be a different tab using ATI). Even if 60Hz is the only thing avaible for you to select in the box then you could try unticking "hide modes this monitor cannot display" since windows may be restricting a setting that your monitor can actually handle. If you do however force a setting that your monitor really cannot handle then you will be presented with a black screen, but don't worry. Don't push anything for 15 seconds and windows will change back the settings to the last settings that worked. No problems.
Now as to how refresh rate affects your game.
While it may seem to be fairly easy to change your refresh rate in windows unfortunately these settings do not apply in games. This means that 99% of games will always run in 60Hz regardless of what you set in windows. The flicker problem that I mentioned (with low refresh rates), even if you were able to see it with a white screen, is much less obvious when you try and see it in darker colours. Rarely are there large portions of white in a game so in all honesty refresh rate does not really become annoying in a game. I will state however that if the refresh rate in a game is 60hz, the highest frame rate that your monitor will actually display is 60, regardless that your computer may be telling you your frame is 90 or 150. There are programs which you can run (such as NVRefresh Tool) which can lock the refresh rate inside games so that you can set it to something other than 60hz. By using such a program you would be able to get a decent refresh rate in games (such as 75 or 85) however as I said refresh rate in games is not nearly as big an issue as it is while using the windows environment.
There are some problems tho, I've noticed compatibility issues when trying to run load a few games, battlefield 1942 for instance wouldn't load properly when I had the NVRefresh Tool enabled. On a side note I would like to point out that in the latest update for Battlefield (v1.6) has actually made Battlefield 1942 the only game I can think of that actually allows you to set the refresh rate in the game (as well as being able to set the resolution). A good move on EA's part.
So finally if you want to learn a bit more about how the refresh rate works just try experimenting with it. As I said, 60 may be the maximum your monitor can handle at your nice insanely large resolution of 2048x1536. If you really wanna see what the refresh rate does you might wanna kick the resolution down to 1280×960 or something else a bit less than what you've got and try seeing the difference between a refresh rate of 60hz and a refresh rate of 85hz. Try looking for any signs of flickering.
Anyways I hope this has helped you a bit.
Patrick
