Magento provides you to make your own store, Facebook integration with Magento increase the publicity and number of users of your store and can add new fb plugins to your store.
So to implement it follow the steps:
Create your own Magento store account in Facebook. To create your stores account refer the link below.
Create a Facebook app. The fb app is the core part which helps in integrating Magento with Facebook using which you can add a number of fb plugins to your store. To create a fb app refer the link below.
Once you are done with your fb app, it provides you with an appID and secretkey which you have to refer in the core files of your Magento store in order to establish a successful connection.
When you got your appID and the secretkey you have to assign the newly created fb app to your store by giving your store url in the “Page Tab” of your facebook app page.
Now you have to go for add Page tab from your fb store account and add your app to your fb store account.
Once you make a integration now you can access the data in the facebook or its plugins but it is required that the client application must also do authentication in order to access the data. For that you have to include methods which present in PHP SDK Library, which you have to download from link below and include to your application module.
Once its done you can add Magento extension module for fb like button, comment box or like box any plugin you want to show on your page.
You can download some from following link to add like button.