J2EE is the "Enterprise Edition" which adds a framework for server-side web apps. I don't remember if JDBC (the database interface code) is part of J2EE or a separate package. But it's for making online stores and other useful nonsense that usually use a database server for "persistence" of objects. See this: http://download.oracle.com/javaee/6/api/
Mobile devices have a sub/extended-set called JME, but I don't know how that now relates to Android and the bouhaha with Google.
Mostly you need to go to the great-satan-Oracle's java site and see if you can make any sense of today's plethora of offerings.
There are several versions of java:
standard edition (SE): the "core" java if you will
enterprise edition (EE): things like web apps and lots of enterprisy goodies
mobile edition (ME): for mobile devices
If you want to learn java you will have to start with standard edition (likely what you refer to as "core" java) as this will provide you with the basics of the language.