I am looking for an opensource API for my existing website. The functionality that I am looking for is:
What are my options?
All the things you ask for can be done by any tool that analyses server logs apart from the geographic location thing. (I haven't seen IP addresses coming out of Google Analytics, so I wouldn't recommend that for your requirements, though I'd recommend it generally.)
The awstats tool is widely used and has a plugin for geographic location so I'd recommend that.
Does it have to be open source or just free? If it's the latter use Google Analytics.
As other people have said if you are concerned most about free software vs open source Google Analytics gives you everything except IP addresses. The best solution may be for you to use Google Analytics and something like awstats.
By combining the two you will get everything you need and have two sources. Which is usually a benefit because of the complexity of data collection. Sometimes one Analytics tool will misreport numbers or show something confusing and having two sources of data will allow you to compare and hash out slight differences that can give you an edge in the long run.
You might want to take a look at piwik. There is a plugin for geolocation.
You will have to install it in you server.
I didn't exactly understood what you mean by API, but Piwik is simple and reliable. Try also Open Web Analytics, much more sophisticated than Piwik, but a little buggier and not as easy to use.
