This is the "point and shoot" automatic mode. The camera will take care of all settings in this mode. While this is fine for most settings, flash is not one of them. Thus, shooting in auto mode, you may become tired quite quickly with the flash popping up automatically at the worst moment possible to ruin many shots.
This is the same as automatic mode but with flash off. So it offers basically the same convenience of auto mode but without the occasional nasty surprise of the flash popping up when it feels like it. If you think you are an absolute beginner, you can use this mode pretty much all the time.
I assume that anybody who has ever picked up a camera can decide for themselves if they need flash in a shot or not. So, if you need to use the flash, press the flash button to let it pop open, set auto mode and flash away. If you don't want to use the flash, stick to flash-off mode.
Portrait mode will set the auto-focus on the closest subject and soften or blur the background using a large aperture. It will also play with optimizing the image to produce soft skin tones. White balance and metering are also set automatically.
Landscape mode tries to give you as much of the scene in focus as possible using a small aperture. Flash and the auto-focus assist are disabled.
Child mode... no idea...
Sports mode will set a high shutter speed to allow you to freeze the action. The autofocus will work continuously as long as the shutter is held half pressed.
Macro mode allows you to take close up shots of objects. The flash will pop up as needed, and the camera will focus on the center auto focus point.
Night potrait mode tries to strike a balance between the subject and the background in low light. The flash will pop up automatically if needed. Shutter speed can be quite slow so take care about camera shake.
