The awards that show up under your avvy are medals given out for winning one of the forum's community competitions, for making a long lasting contribution to the forum or for taking part in one off events. Here is a list of medals. You can also hover over the medals under people's avatars to see what they're for.
The awards you're most likely being notified of getting are Trophy Points. These are earned by completing various tasks around the forum and do not have any visible reward. The accumulated points do show up though in your postbit (underneath the amount of messages you've posted). Here you can find a list of what requirements need to be met to unlock more points.