Badges are fun reward for participating in the community. The more you participate the more badges you will earn. Many of the badges are part of a series, whereas you take that action more you work up the badge levels.
Where to find your badges
From your profile, your two most recent badges will show here, with a view all badges link.
Click on the view all badges link to see the full badge page. Badges you have earned will be solid color on your badge page, those still to earn will be greyed out until earned. Click on each badge to see the action required to earn it.
You can view other community members badges from their profile.
Other places in the community you will see your badges: in your notifications page, as you earn new badges, they will appear here.