How to create a jQuery function (a new jQuery method or plugin)?

Yup — what you’re describing is a jQuery plugin.

To write a jQuery plugin, you create a function in JavaScript, and assign it to a property on the object jQuery.fn.

E.g.

jQuery.fn.myfunction = function(param) {
    // Some code
}

Within your plugin function, the this keyword is set to the jQuery object on which your plugin was invoked. So, when you do:

$('#my_div').myfunction()

Then this inside myfunction will be set to the jQuery object returned by $('#my_div').

See http://docs.jquery.com/Plugins/Authoring for the full story.