The main issue is that Google is using Java syntax, but they are not
using the Java VM itself. They have their own implementation called Dalvik. Instead of licensing Java, they copied the Java APIs and essentially made an "imitation" version of Java.
Imagine if you built a car that looks very similar to an existing make and model, and you gave it the same name as the existing one. Obviously the manufacturer of the original one isn't going to be too happy
Do you have to have a license to use java?
Not if you're a regular user, probably if you want to make money off your own implementation of it.
Is this like microsofts .net framework in that you need one (which I also don't use).
The .NET Framework and a lot of the core development tools actually comes bundled with Windows (and newer versions available as a free download). C# and bits of the .NET Framework are both open (ECMA) standards, whereas Java is not. Microsoft has stated that they have no problem with the Mono project (that lets you run .NET code on Linux or Mac OS), and in fact, Microsoft helped Novell with Moonlight (implementation of Silverlight for Mono)
There's also a legally binding promise from Microsoft
to not sue anyone for creating their own .NET implementation. So if Google used C# and .NET instead of Java, there'd be no problems right now.
Repeat after me: jQuery is not