An API is a description of a means of communicating or interfacing between 2 otherwise separate systems. Systems in this case can vary in scale and scope, but the common use case is between some computer code that you wrote and some library of computer code that someone else wrote.
A good way of looking at this in the material world is to think of an API like the electrical socket in the wall. It has a particular number of pins, shaped, spaced and oriented in a particular way. For those who haven't travelled you might be surprised to know that there are currently 15 different types of sockets used world wide. Here are 14 of them (didn't dig to find out which one is missing).
Ie each type of socket corresponds to an API
Devices with leads ending in a particular plug only fit into (and work with) sockets with the matching configuration. And that's how 2 separate systems work with each other. In the example above, system A is the power network supplying power into my house, system B is any device that wants to draw on that power.
Oracle (Sun Microsystems actually - Oracle just has deep pockets) came along and created the Java language along with a bunch of standard code libraries that anyone using Java could make use of. In order for you to use these libraries they needed to have an API, otherwise you couldn't talk to them.
This is the same as Oracle laying power cable to your door and providing electrical sockets for you to plug your devices into. In this analogy, Oracle created an environment in which there was a reason/market for others to create devices to plug into their electricity network. And they did, big time. There are millions of apps written in Java, just like there are millions of different devices to plug into the wall socket.
Then Google came along, laid their own power cable to everyone's door, provided their own power and <shock/><horror/> they used the same socket layout as that originally used by Oracle. To follow the analogy, they did this so that anyone with an existing device could choose to plug it into either electricity network. If Google didn't do that you would need to have 2 kettles, one with each type of plug, 2 fridges, 2 TVs (well many of you already have that) or at least have 2 adapters for every device (but then you have just pushed the problem back to the socket on the device), OR you have to lock yourself into one network or the other.
Oracle's claim on the API infers ownership over any device created by others that uses that API (This is the network lock in mentioned above). Why would your power company have any ownership of the kettle you plug into the socket?
This is especially pertinent to code libraries written by others (and there are millions of them) as they all have to use low level building blocks in order to do things. Oracle provided a description of those standard code libraries and the first implementation. Others (Google is but one) have come along later and provided their own implementation. Because everyone is using the same APIs a developer can choose the libraries they require and the implementation on which they want to run.
This leads to large amounts of innovation as everyone can choose the most appropriate components while still owning their own creations.
As an indie software developer, for Oracle to think that code that I have created is in some way owned by them and that I can't take my own code and run it elsewhere is outrageous. And is the total antithesis of co-operative competition.
There is more to this argument, but that is it in a nutshell.
