1600X1200 is an old 4:3 display standard. New displays are mostly 16:9 or 16:10, though we have some exotic displays coming out now that are much wider.
Basically, the more pixels, the better. 1920X1080 is the standard and 16:9 ratio. I happen to prefer the 16:10 display, 1920X1200, I like that extra height, better for reading text.
For the best gaming, the large 27 and 30 inch displays are really nice, 2560X1440 (16:9 27 inch) and 2560X1600 (16:10 30 inch). Some like to go even further with a triple display setup (note, don't go with two displays for gaming for you end up with a black bar right in the middle of your view).
The problem with the bigger/multiple displays is cost. Not only to the displays cost more, but you need at least twice the graphics processing power and RAM to run them. I recently bought a nice 30 inch display, 2650X1600, and am using a GTX Titan to drive it.