How do I implement an application in which user can be either authenticated by third party or by my own application?

You have to separate a few things and think about what /how you want to achieve authentication and authorization. User Interface - Web pages -  You have  both Social authentication  and your own.  When you use your own authentication, what you are doing is , authenticate ( verify username and hashed password against your database or whatever repository and framework you are using ..Spring + LDAP) first and then create some form of token and attach that to cookie /session, etc. ( Mostly its taken care by framework but you get the idea). When you use social authentication, after you re-direct the user to authenticate against http://Facebook.com  they are redirected back to your app, once you verify /acquire the access token ( its all part of Facebook OAuth integration , they have samples in many languages), you have to follow the second part of "your own authentication"  , the on I underlined above.  If you check the framework you are using or ask around for that framework if they support Facebook login, then you know it can be easily integrated and then you can extend that to provide other 3rd party logins ( Twitter , google, etc.)   Now comes the REST API.  Authenticating the user for web app is different than accessing the REST API. When you access the REST API, you are doing this behind an app ( Web App, Mobile app , etc.) .  Yes you can easily enable Basic Auth but then you have to not only give away username and password to every app but also make sure they use it properly ( like not storing them on JavaScript and sending the username and password in REST API call from browser). So what you need is, a way to obtain OAuth token  ( or another token /standard) after user is authenticated and use that token to access the REST API.   IF I understand it correctly you are not looking to expose your API for 3rd party but rather you want to allow 3rd party sign in (Social Auth) but after that user should be able to access UI or REST API.   You can implement OAuth if Its supported by framework, or your own token based system. See my other answer on difference between OAuth and Non authn -   Good luck.

You can check this blog, in which https://scotch.io/author/chrishttps://twitter.com/sevilayha)  explains about the JSON Web Tokens (JWT). They are standard way of implementing authentications. Here is the link: https://scotch.io/tutorials/the-anatomy-of-a-json-web-token

You can use JSON web Tokens for Authentication. http://techarena51.com/index.php/json-web-token-authentication-with-flask-and-angularjs/