The "video mode" settings are used in conjunction with the subsequent "i/p Scaler", "Resolution", "Progressive Mode" and "Aspect" video processing options.
When "video mode" is set to "Game", most of the video processing (if any) is disabled in order to minimize the amount of delay between the video signal coming in from the game console (or PC) and the video output leaving the receiver. This minimizes the "lag" which sometimes causes problems for gamers. "Game" would be appropriate if you have a gaming console connected to the port you're configuring. "Movie" probably would be appropriate if a Blu-ray player is connected. "Auto" probably would be appropriate if you're using a PlayStation3 for both games and movies. In most cases, "Auto" should be fine, though.
The "i/p scaler" setting is used in conjunction with the subsequent "Resolution", "Progressive Mode" and "Aspect" settings. Note that scaling is not available when an HDMI input video signal has too high a bitrate, e.g. when x.v.Color (deep color) is enabled. (The text doesn't explicitly say "too high", but all of the options it lists cause increased bitrates.) The "i/p scaler" setting enables (or disables) conversion of the video signal's resolution for analog input, HDMI input or both types of input connections. "Off" usually is reasonable, so that the input signal is passed unchanged through the receiver to the TV (or projector), so that the TV itself scales the video signal to match the native resolution of its display.
Sometimes, though, it's appropriate to scale the video to a particular resolution before it gets to the TV, often because the TV's scaler doesn't do as good a job as the receiver's scaler does. In that case, you'd need to know the exact resolution of the TV's display, so that the signal doesn't wind up being scaled twice (first in the receiver and then again in the TV) and made to look even worse.
The subsequent "Resolution" setting determines the exact scaling to be used for the output signal if the scaler is not "Off". "Auto" usually is reasonable. That causes the scaling to match the capability that the display device (TV or projector) reports over HDMI to the receiver. If you have an HD TV that's capable of only 720p, for example, with "Auto" set the receiver should automatically scale whatever input signal comes in to 720p, whether up from 480i standard definition or down from 1080p HD. Alternatively, you can set the receiver to output one of the list of specific fixed resolutions.
Similarly, it mght be appropriate to set "progressive mode" and "aspect" if the TV isn't doing what you want.
In most cases, though, it's appropriate to have other devices do the scaling-- either the source device (BD player, cable box, game console, etc) or the TV -- and not the receiver, so I recommend setting "Video Mode" to "Auto" and "i/p Scaler" to Off.
Does this clarify things at all?