So first of all, what is the difference betwen a programming library and a programming language?
A programming language is a means by which we (as humans) can give instructions to a computer to get it to do various tasks (algorithms, computations, etc...). They are largely artificial constructs created for human readability which then need to be compiled or interpreted into actual instructions which the computer can then carry out. A compiler turns a program into machine readable instructions ahead of time (compile time) where as an interpreter does so on the fly (runtime). Languages such as C and Java are generally compiled and JavaScript, Perl and Ruby are generally interpreted. While there is no real reason why this has to be the case, these factors normally affect the syntax of the languages in question.
A library is a collection of functions and methods of (normally) common actions which are not expressed in the original language. These are not included in the main language because not everyone will need to use them all the time. They are also written independently of the main language compiler/interpreter and they also encourage code re-use (so not everyone needs to write their own String.trim() function for example).
Because of the fact that there are many JavaScript interpreters out in the internet and JavaScript developers cannot control which one will be used to interpret their code, a few years ago some people released their own framework/libraries which iron out these differences and allow developers to write one set of unified code from which any browser differences could be abstracted out into the library.
jQuery primarily gives developers a generic way to access and manipulate DOM elements (one of the main differences between browsers).
However, the syntax used looks very different from plain old JavaScript.
Here is a typical JavaScript loop.
var elements = getElementsByTagName("div");
for (var i = 0; i < elements.length; i++){
alert(elements[i].id);
}
Here is the same loop in jQuery
var elements = $("div");
elements.each(function(i){
alert($(this).attr("id"));
});
Do these look remotely alike?
How about this?
JavaScript
var element = document.getElementById("myDiv");
alert(element.innerHTML);
jQuery
var element = $("#myDiv");
alert(element.html());
To the untrained eye, one might think these are different languages.
jQuery is not another language though, not only does it use the same JavaScript interpreter, not only is it useless without JavaScript (i.e. it is not independent), but in jQuery you are really creating and manipulating a jQuery object and you are given the option to access the JavaScript object it references using the get() method.
It is possible though to write a language based on another language. The Ruby runtime is essentially built on the C++ framework. JavaScript runs in browsers written in C++ as well. These languages have no direct relation to C++ in their syntax though. In theory you could probably write a C++ compiler in Ruby should you want to (though that would be pointless and slow). You can write programs which translate programs from one language to another (the Google Web Toolkit translates Java into JavaScript).
I would argue that jQuery is a dialect of JavaScript or else even a kind of JavaScript slang. It differs from the main language, but it doesn't veer far enough to be completely independent or incomprehensible.