Use of AF button and AF Mode in Canon DSLR?

Auto Mode will use Autofocus if it's enabled (as will every other mode), but other than that, they're completely unrelated. If you want to focus manually, turn the AF/MF switch to MF, especially if the lens(es) you have doesn't allow for manual focus while AF is on (some do, some don't - read your manuals). If you want to not have to worry about shutter speed, aperture, ISO, white balance, etc., use Auto Mode, otherwise, if you want some level of control over one or more of those, pick the appropriate mode - again, read your manuals to know which modes control what parameters.

The "Auto" mode on the camera's Mode Dial is primarily for Auto Exposure, though it does also restrict what choices you may have available with regard to autofocus.

• If the lens switch is set to MF (Manual Focus), setting the camera's mode dial to "Auto" will not override that. The lens will not autofocus and you will need to focus it manually.
• If the lens switch is set to AF (Auto Focus), setting the camera's mode dial to "Auto" will allow the camera to use autofocus in the way it thinks best, rather than letting you choose other ways of using autofocus, such as manually selecting the AF point or zone, or manually selecting the AF Mode (One Shot, AI Servo AF, etc.).