Java Script was created by Netscape for their new version of browser. Initially they released this as part of Netscape Navigator 2.0 with a name "LiveScript". Later they renamed it with "Java Script". Note, Java is a programming language and completely differ. Actually at that time Java was being most popular.
In the race of Browsers, Microsoft was chasing Netscape and so it invented 2 new scripting languages: 1) VBScript and 2) JScript. (with more commands and features like access to ActiveX which were not present in Netscape's JavaScript)
The issue in developing web application is they run on different browsers and we don't know on which browser end user will access. All browsers are not same in features and commands. Cross Browser Compatibility issue arise in use of Scripting languages whether it is JScript or JavaScript. jQuery take the role for handling this issue.
John Resig's (creater & lead developer) jQuery is an Open Source, consist library of functions written in Java Script. Day by day no of functions are increasing as it is open source. Developers just needs to know how to use and it is easy to learn. It also promote code reuse. It also supports JSON for transporting data from server to browsers nativelly. No need to write complex JavaScript functions instead jQuery is simple and short. jQuery commands can be chained together. jQuery is more powerful and it lets you to also manipulate CSS. Its library file can be installed in project by downloading it from http://blog.jquery.com/2011/03/31/jquery-152-released/ . The jQuery library file can be used without installing it in our project by giving the reference of jQuery, Google and Microsoft http://docs.jquery.com/Downloading_jQuery#CDN_Hosted_jQuery