Generally speaking, response time is the amount of time it takes the pixels to changes colors; from one shade of grey to another. The faster the response time (i.e. the lower it is) the less likely you are going to ghosting or image after effects. But, response time is not an exact science. Manufacturers simply measures the fastest time and discards the slowest times. If monitor A's fastest response time is 2ms, but it's slowest response time is 300ms, then it is simply rated at 2ms. If monitor B's fastest response time is 5ms (worse than monitor A), but it's slowest is just 250ms (better than monitor A), then it is simply rated at 5ms.
Arguably, there is not much difference between 2ms and 5ms. Some will even say that the monitor's response time can be as slow as 12ms and you still will not detect ghosting. I would say you should not buy anything that is worse than 6ms; good thing a lot of monitors have 6ms or less response times.
ssddx brought up input lag. Input lag is basically the amount of time it takes for your actions to be displayed on the screen. If you are playing a game like Crysis and you press the mouse button to fire you gun, but you have to wait one second for it to actually happen in the game, then the input lag is one second (it's actually never that high). Input lag is basically the time it takes for the electronic signal to travel from the mouse and keyboard to CPU to process that instruction, then for the video card to send that signal to the monitor's (or HDTV's) input port and then travel through the various electronics within the monitor and finally displayed on the screen. I skipped a lot of steps, but you get the idea. There will always be some input lag, it cannot be avoided. However, it can be minimized by choosing the correct HDTV or monitor. How do you know which HDTV or monitor has the lowest input lag? Someone must test it.
The reason why you want to set a 120Hz or 240Hz HDTV to 60Hz or "Game Mode" is to reduce artificial input lag. All HDTVs only accepts 60Hz (or 60 frames per second), however they can output 60 frames, 120 frames (120Hz) or up to a "simulated 240 frames" (240Hz). How is this done; taking in only 60 frames per second, but outputting 120 frames per second? It's done through video interpolation. In between every two actual frame, the HDTV's electronics creates an interpolated image. This helps smooth out video playback and can also draw out details. Great, but it takes time to do it, thus you get input lag.
The same thing with 240Hz mode, but there are two different 240Hz modes; "real" and "fake". "Real" 240Hz mode does even more processing so that the input lag is even greater. According to a friend of mine who as a Sony 240Hz HDTV (I don't know which model) the 240Hz gives a little bit of depth to the picture. It's kinda like you have a picture of kids playing in a playground that's made up of layers of paper on top of each other. The background of the park is a sheet of paper, the kids playing in the background is a cut out that is placed on top of the background, and finally there is a paper cutout of you in the foreground and that is placed over the other kids. "Fake" 240Hz is simply 120Hz with the HDTV's backlight strobing on and of very quickly, so fast you don't even know it. What the hell does strobing has to do with anything? I'm sure you've been to a dance club. Sometimes they turn off most of the light and then they have strobe lights flashing on and off. What do you see? You see other people around you for a fraction of a second that seems to be have been captured like in a picture very sharply. That's "fake" 240Hz mode. The strobing effect is supposed help sharpen video on the HDTV does? As an owner of a "fake" 240Hz HDTV... I don't think so.
Regarding multi-player online games... A lot of the lag is due to "ping" which is the amount of time it takes for data from your PC to travel thru the internet to reach the server that is hosting your game. Most of this type of lag is out of your control and sometimes it can be very little lag, other times there could be a lot of lag. Some internet service providers have more ping than others. Solution? Switch to an ISP with less ping. If there isn't a better one, then... too bad. The distance between you and the server that's host the game also plays a role. You want to choose the closest server. Then there's internet traffic... sorry but there's no way for to force other people off the internet... unless they are on the same home network you are and you control the router. Back to the game hosting server... if it is being overwhelmed by the number of people connected to then you have a problem. I'm sure there are other issues related to network lag, but I don't play online games so I never researched it.
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