I'm new to Android developing, and I would like to know what is the connection or difference between the version and the API level?
What is each one referring to?
And what does it mean when I decide to develop some app for API 14 or for Android version 4.0?
Is one a subset of the other? I simply don't get the difference, and why are there two counters?
Well, API is for development, so the changes in new API version are more "inside". But new version of Android usually adds more features for users, that are "visible".
Check this page http://developer.android.com/guide/appendix/api-levels.html, there is a table that shows relations between versions and API levels.