It is very wide question and have so many answers and techniques based on your requirements, for example if you need to implement it yourself you need to use token based authentication, AKA JSON Web Token, technique where you simply send a token to you client after validating their username/pass to be saved in the browser and later on you can send this token along with any further API/HTTP request(s) for validating your encoded token that was sent before by your authentication service.
Check JWT site to know more about different libraries that will help you implementing this technique in almost all known languages (PHP, NodejJ, Python, .NET, and Java) in just 10 minutes.
Backend server or used language should not be the issue as much as you should understand the technique itself.
If you want to use Amazon Cloud, you can totally depend on their authentication services they provide, its name is AWS Cognito
If you need to implement social based authentication, then you should use OAuth protocol for integration with different sites like Facebook/Twitter