how can we make our own site API. just like facebook provide API and we can connect to the facebook
Assuming you want an api for your site LIKE facebook has an api for their site. Make a web service for you website. So if you go to "yourwebsite.com/something/something/" it can return a XML file that people can look at and use. Then you could make a wrapper for a certain language, so people can access your web service via that language.
This is about as simple as it gets: http://support.microsoft.com/kb/308359
Just google search "Web Service how to" and should find some good tutorials.
The Facebook API is mostly just a set of URLs on their web site that can be called through HTTP from other servers to obtain documented results.
All you have to do is code your server to respond to requests on certain URLs, write the documentation, and call it an API.
Writing a good API, on the other hand, is harder (but we'd have to know what you're trying to do). For instance, the part about where you can use Facebook to connect to third party applications is a known technique generally mentioned as SSO.
@user438860 "how can we make our own site API. just like facebook provide API and we can connect to the facebook"
How can you make your site API? With software engineering.