Facebook is a social media web site that has become very popular since it's establishment in 2004. It's main purpose is to connect "friends" through a network, enabling them to share thoughts and images with each other and other friends in their specific group.
Both the PS4's and PS3's PlayStation Network as well as the Xbox One support connecting to Facebook and Twitter in order to post status update (like Trophy or Achievement gained), screenshots, and videos. (Note that not all of these features are available on all of these consoles, and for some of these one has to subscribe to a paid service like Xbox Gold.)
Nowadays many games, among them some Tomb Raider and Lara Croft Games, feature a connection to Facebook in order to share score information or find other players playing the same game. This is a part of so-called Social Gaming, which aims at bringing the players closer together and engage them in competition.
Lara Croft: Relic Run
Connecting to Facebook in the Mobile game Lara Croft: Relic Run for the first time gives the player 25 Gems. Furthermore after connecting the player can compare his or her score directly on the map and they can send Curses to one another's friends.
Many people refuse using these Social Gaming features or services like Facebook at all because of the fear that their private data and usage profiles will be used for personalised advertisement or other, more sinister purposes. Fact is that by creating a user account at Facebook, Twitter, Google, or other social media web sites one relinquishes many rights to one's personal data that can and will be used for such things as advertisement or user tracking. Even if not actively participating in these networks users may be tracked via the social media buttons nowadays embedded in many web pages.
On the other hand the user "pays" for the services provided by these networks with their user data: The collected information is used to pay for maintenance and infrastructure. This way the service can stay free of charge.