If you go out to a bar for example, you tell Foursquare where you are ("check-in"). To do this, it uses GPS to give you a listing of nearby places from which you select your location. It will then notify your friends of where you are.

The 'game' component comes from a points system used to encourage you to go out and explore. You receive points for each check-in. Bonus points are on offer for those who are the first to register a new place. You can even become Mayor by being the person to have visited a particular location the highest number of times.
Foursquare integrates with other web 2.0 tools. If you choose to you can link your Foursquare account to your Twitter and Facebook accounts and allow it to report updates. Foursquare itself keeps stats on you - they track your activity - number of nights out, number of checkins, ave. no of checkins per night out, distribution by day etc...
Foursquare seems to have found its niche by creating partnerships with businesses that produce shopping and restaurant guides. This gives them a way of creating a money-making aspect to their resource. There is an interesting article about this on TechCrunch.
Yelp is another social networking site that enables sharing of reviews for local businesses, restaurants, shops and the like, but it appears to focus more strongly on the review aspect than does Foursquare. Yelp also has not been around as long as Foursquare and although it seems good it has a more limited geographic representation.