At bird's eye view, Java EE is a platform, i.e. something that we can build on.
Taking a more technical perspective, the Java Enterprise Edition standard defines a set of APIs commonly used for building enterprise applications. These APIs are implemented by application servers - and yes, different application servers are at liberty to use different implementations of the Java EE APIs.
However, the java ee library will not work, nor compile unless if your code is being run on or has access to a Java EE application server (such as JBoss, GlassFish, Tomcat, etc).
You compile against the Java EE APIs, so you only need those APIs at compile time. At runtime, you'll also need an implementation of these APIs, i.e. an application server.
Why do I need something massive as JBoss just to compile simple code to send an email?
You don't. However, if you wish to use the Java EE API for sending mail, you will need an implementation of that API at runtime. This can be provided by an application server, or by provided by a stand alone library you add to your classpath.
Why are Java EE libraries not "standard" and included in the regular JVM download and/or the SDK?
Because only the APIs are standardized, not the implementations.
Why are there so many Java EE offerings
Because people disagree on the right way to implement certain features. Because different vendors compete for market share.
What can one do with Java EE that they cannot do with standard Java?
Since Java EE implementations are built with "standard Java": Nothing. However, leveraging the existing libraries can save a great deal of effort if you are solving typical enterprise problems, and using a standardized API can prevent vendor lock-in.
What can one do with standard Java that they cannot do with Java EE?
Nothing, since Java EE includes Java SE.
When does a developer decide they "need" Java EE? When does a developer decide they do not need Java EE?
Generally speaking, the Java EE APIs solve typical, recurring problems in enterprise computing. If you have such problems, it usually makes sense to use the standard solutions - but if you have different problems, different solutions may be called for. For instance, if you need to talk to a relational database, you should consider using JPA. But if you don't need a relational database, JPA won't help you.
