How can jQuery behave like an object and a function?

Functions are also objects, so $.each can be defined in a similar way as an Object.

JavaScript is a prototypical language. For jQuery, this means that every instance of $ inherits methods from jQuery.prototype.^{See Notes}

A very rough demo, to achieve the similar behaviour:

(function() { // Closure to not leak local variables to the global scope
    function f(a, b) {
        //Do something
    }
    // Prototype. All properties of f.prototype are inherited by instances of f.
    // An instance of f can be obtained by:    new f, new f(), Object.create(f)
    f.prototype.removeClass = function(a) {
        return a;
    };
    function $(a, b) {
        return new f(a, b); // <--- "new f" !  
    } 
    $.each = function(a) {
        alert(a);             
    };
    window.$ = $; // Publish public methods
})();

//Tests (these do not represent jQuery methods):
$.each("Foo");                   // Alerts "Foo" (alert defined at $.each)
alert($().removeClass('Blabla'));// Alerts "Blabla"

Notes

jQuery’s root method is defined as follows (only relevants parts are shown):

(function(win) {
    var jQuery = function (selector, context) {
        return new jQuery.fn.init(selector, context, rootjQuery);
    };
    //$.fn = jQuery.fn is a shorthand for defining "jQuery plugins".
    jQuery.fn = jQuery.prototype = {
        constructor: jQuery,
        init: function( /* ..parameters.. */ ) { 
            //.... sets default properties...
        }
        //....other methods, such as size, get, etc...
        //.... other properties, such as selector, length, etc...
    };
    jQuery.fn.removeClass = function() { // (Actually via jQuery.fn.extend)
        // ... method logic...
    };  //...lots of other stuff...

    win.$ = win.jQuery = jQuery; //Publish method
})(window);

The advantage of the prototype method is that it’s very easy to chain methods and properties. For example:

$("body").find("div:first").addClass("foo");

A method to implement this feature could be:

$.fn.find = function(selector) {
    ...
    return $(...);
};

If you’re interested in jQuery’s real implementation, have a look at the annotated source code:

• jQuery core – Definitions of the constructor and base methods.
• jQuery.fn.extend is used to add removeClass, etc. to jQuery.
• jQuery 1.7.1.