By core java, I am assuming you meant Java. Java is a programming language. Javascript is a scripting language.

Javascript requries a web browser or similar to function.

Java and Javascript has confused people a lot due to their naming similarity.
But remember, I repeat remember, Java and Javascript have no connection at all. They are completely different technologies.
Java is a programming language used for creating standalone applications as well as applets in browser.
Javascript on the other hand is a scripting language designed specifically for browsers. All modern browsers have built-in support for javascript.

Yes, JavaScript is more on designing and for Website Developing.

Java is an object oriented programming language developed by Sun Microsystems that is capable of running on multiple operating systems with the use of an interpreter.
JavaScript, on the other hand, was created by Netscape as a scripting language much like a client side version PHP. It cannot create stand alone applications, but instead resides on an internet browser.

Java was first,

Then there was a scripting language invented called ecma script.

They wanted to make it popular so stole the 'java' part because 'ecma' wasn't cool enough.
ECMAScript - Wikipedia, the free encyclopedia

Damn Internet marketers making it confusing for everybody........

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Core java is all the basic classes and API's. It is the Java equivalent
of the "C lib" or the intrinsic controls and built-in-functions in VB.

These are then added to with extra API's to make a "platform", that is
the complete set of API's needed to run an application.

Core java + common API's exist for different application areas, so we
have the J2EE platform the J2SE platform and the J2ME, for example.

Hmm.. I got really nice answers, Thanks so much friends for helping me

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Java is an object oriented programming language that is capable of running on multiple operating systems with the use of an interpreter.

JavaScript, is a scripting language much like a client side version PHP. It cannot create stand alone applications, but instead resides on an internet browser.