How to add a Twitter feed to a website without using a standard widget?

Question

Twitter

How to add a Twitter feed (any user) to the site without using the standard widget? Is it possible to receive tweets by entering only the username or user-id? For example, on the KinoPoisk website, a Twitter feed has been added to almost every actor. How do they get messages from twitter?
PS And how much will it cost approximately if you order this to be implemented?
I hope I made the question clear. If not, then correct me =).