foursquare is a location-based social-networking service that issues badges and points to users who "check-in" to a location or place of business.
The act of checking-in is where the user, with their GPS enabled device (such as a smartphone), reports to the service (and by extension their friends and contacts) on the service of where they are.
Local businesses and companies can also leverage this information to offer discounts and services.
If a user checks into a place more than other users of the service within a 60 day period, and they meet the time and distance requirements as well as having a profile photo, they are crowned the "Mayor" of that location. They will hold that title until another foursquare user checks in more times than the previous mayor.