The mail delivery mini-games are all different. During them, you will have various abilities. The two most common are toss mail and toss bone.

click to enlarge
Tossing mail will throw the mail at its intended recipient, or at their post box. You must target the box by clicking on it before tossing the mail in order to deliver it.
Tossing bones drops a dog bone at your feet. This is used to distract dogs when gathering or delivering mail; for a few seconds, they will chase down the bone instead of you. If a dog does chase you and comes in contact, they will knock you down, stunning you for several seconds.
You will also be asked to gather mail many times. Mail bundles will appear as green dots on your mini-map, and will show with bright sparkles when you're near. Simply click on them to pick them up.