You mainly want low input lag, low pixel response time, and if you play games that get dark in places (any that have a gamma or color test screen definitely count), good matching to sRGB (a computer/TV-oriented color space that took into account typical home and office lighting, and limitations of cheap CRTs, which LCDs can now easily match or best).
Most PC monitors today either "just are" well matched to sRGB, or have a decent sRGB preset.
Fast pixel response time reduces/removes ghosting and smearing of the image.
Input lag refers to the time between the GPU sending the buffer's contents to the monitor, and the monitor putting those contents on screen. This being low allows faster/smoother responses in FPS games.
Almost lastly, refresh rate. 120Hz or higher can allow for even lower lag between your mouse clicks and the display. But, that working depends on a monitor with a total response time under a single refresh (~17ms for 60Hz, ~8ms for 120Hz). At larger sizes, there aren't many, if any, capable panels for that, so prices go up and availability down. Even 27" ones are typically above $500, with a few 2560x1440 Korean exceptions.
Finally, there's Freesync (AMD only) and G-Sync (NVidia only) coming down the pipe in some new monitors. G-Sync monitors cost a premium for the feature. These implement adaptive refresh rates on the monitor, so that v-sync can be on with any arbitrary refresh rate, and get any arbitrary effective FPS.
Under $500 will not get you
any 40" monitor, however, at any resolution, buying new.
http://www.hdtvtest.co.uk/news/input-lag
There's one DB, with only a handful of TVs that have low input lag, and the only one that might be within your price range appears to be a European-only variant (I can only find European sites selling it, so no USD prices).
http://www.blurbusters.com/faq/120hz-monitors/
Another.