I'm going to explain about few important packages in core Java.
1. java.lang: This package got primary classes and interfaces essential for Java program. It consists of wrapper classes (wrapper classes can be used to convert ordinary data into objects), Strings, Threads etc.
2. java.util: This package contains useful classes and interfaces like Stack, LinkedList, Arrays, ArrayList, List etc...
3. java.io: This package handles files and input output related tasks.
4. java.awt: This package helps to develop GUI. It consists of an important sub package java.awt.event to handle the events for GUI elements.
5. java.swing: This package helps to develop GUI like java.awt. In fact, this an extension to java.awt.
6. java.net: Client - server programming can be done using this package. This is a very important package to develop any web program like web browse or web server.
7. java.applet: Applets are programs which come from a server into a client and get executed on the client machine. This package was very well known to everyone before the development of servlets.
8. java.sql: This package helps us to connect to database like Oracle.