jQuery is an open-source JavaScript library developed at BarCamp 2006 by John Resig. Its aim is to simplify the interaction between HTML and JavaScript.
Minify in the context of jQuery means that your code is being compressed by jQuery’s compression tool. Basically, this avoids frustration on the user side by decreasing download times for websites. To even further reduce the amount of code the “chainability” of jQuery code is helping: Each of the individual methods return the jQuery object, allowing you to continue applying methods to the current set of elements.