Higher resolution means that the game will have a more detailed picture than a lower resolution, you will also get a wider field of view in higher resolutions. For LCD monitors ideally you would run everything at the monitor's native resolution, typically the maximum resolution it supports. That will give you the best picture quality.
When people say they max out a game, that typically means they have turned all the advanced graphical settings up to maximum, ie. highest texture details, maximum Anistrophic Filtering (this reduces the blurriness of textures that run over longer distances) maximum shadow quality, highest water details, etc. Someone who maxes out a game may not crank the AA to maximum though.
AA or anti-aliasing is a process that smooths out the jagged edges that naturally appear on objects rendered in 3D. This does come at a price though, framerates tend to take a significant hit when AA is enabled, particularly if you crank it up to the maximum level, and depending on how powerful your graphics card is, may reduce the frame rate to unplayable levels when used in conjunction with high detail settings.
The jagged edges are most pronounced at lower resolutions, and if you want to completely smooth out the jagged edges at a lower resoltuion you will have to use a higher AA settings ie. 8x (8 pixels per sample). As resolution increases the jagged edges do get far less pronounced, and the amount of AA required to smooth the edges decreases. For example at 1280x1024 to completely smooth out the jagged edges, you will probably have to use 8x, or 4x at the very least. However at higher resolutions ie. 1920x1080 an AA setting of either 2x or 4x would be more than adequate to smooth out the edges, and anything higher will not offer an improvement in visual quality and only slow down your framerates. Higher resolutions do require more graphics horsepower than lower resolutions, and turning up the AA to maximum at high resolution can push framerates quite low, even with higher end cards in the more demanding games.
For most games you can find these options in the video or graphics menu, though for a lot of console ports you have to go to the advanced video screen to change the graphical settings discussed above, as the regular video screen tends to only give you things like brightness control and screen resolution. Games running on the Unreal engine also tend not to have AA settings due to the engine generally not supporting that feature. To get AA in those games you will have to try to force it with your video drivers.