Though it runs on multiple platforms, there's something to be done specifically on certain platform. One of it is Android OS.
If your Android device has no Facebook app installed, everything is as smooth as silk, but it you has Facebook app installed, your problem comes. There's few helpful videos in Facebook sdk website, however to make it works for almost all circumstances, there's few extra steps to be done. Let's start with basic stuffs first.
1) Go to download the latest Facebook-SDK for Unity, at the time of writing it's 6.2.2.
2) Launch Unity 5 and create a new empty project. Import the Facebook-SDK for Unity ("FacebookSDK-150414_6.2.2.unitypackage")
3) The Facebook-SDK for Unity comes with an interactive scene. Go to "Assets>Examples", open the "InteractiveConsole.unity" scene. This is the place you can test all the Facebook APIs, before integrate it with your game.
4) To run this in Editor, there's few setup you need to do, as describe it the SDK.
5) First, go to "https://developers.facebook.com/apps/" and on top menu, click "My Apps">"Add a New App".
6) Go to "Advanced setup" for familiar the whole picture.
7) Create a new app id.
8) You should be brought to the dashboard of the new app you created. You have "App ID" and "App Secret" given, and you need these info later.
9) Go to "Settings", and key-in your email address.
10) Go to "Status & Review", and make it public. Now these are all the steps to make Facebook SDK run on Unity Editor, but more to come to make it run in Android later.
11) Go back to Unity Editor of the project you created just now. By importing the Facebook-SDK for Unity, you should have a "Facebook" menu right now. Click on it, and you should see something in "Inspector" pane. Copy the "App ID" from step (8) and paste in the Inspector section.
12) Click "Play" and you should see the "Game" section is active by now. Click on the "FB.Init" button.
13) Click the "Login" button, if your AppID is correct. You can see the AppID you pass to Facebook server in the status pane at the top.
14) In Unity Editor, you will be greeted with another pop-up, click "Find Access Token" to get a valid "User Access Token". Don't worry, this screen will be replaced with Facebook login in actual device.
15) It will trigger you default OS browser, and see it supplies along with "AppID" again.
16) For the first-time, you will need to grant permission to gain access. Click that hyperlink in the "User Token" section.
17) Then it'll tell you the token is created. Click "Continue".
18) You will be brought back to previous page, but this time with "User Access Token". Copy this "User Token" value.
19) Paste it in the Unity Editor dialog in step (14) just now.
20) More button will be revealed in the Editor, if the token is valid. Now you are official logged-in. Feel free to try other function.
We proceed to test on Android device testing in another page.