Update 2015

There are classes in JavaScript they just aren't used on older browsers:

https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Classes

It has constructors, extensions, and the like.

class Cat { 
  constructor(name) {
    this.name = name;
  }
  
  speak() {
    console.log(this.name + ' makes a noise.');
  }
}

class Lion extends Cat {
  speak() {
    super.speak();
    console.log(this.name + ' roars.');
  }
}

A Class in JS:

function Animal(){  

    // Private property
    var alive=true;

    // Private method
    function fight(){ //... }   

    // Public method which can access private variables
    this.isAlive = function() { return alive; } 

    // Public property
    this.name = "Joe";
}

// Public method
Animal.prototype.play = function() { alert("Bow wow!"); }

// .. and so on

Now when you create its object:

var obj = new Animal();

You can expect anything of this object as you would from objects in other language. Just the efforts to achieve it was a bit different. You should also be looking at inheritance in JS.

Getting back to your question, I'll reword it as:

//Class  : A representation of a set with common properties.
//object : One from the set with the same properties.


var Class = function() {alert('bar');}; // A set of function which alert 'bar'
var object = new Class();               // One of the functions who alert's 'bar'.

JavaScript does not have classes, and functions are actually objects in JavaScript (first-class citizens). The only difference that function objects have is that they are callable.

function func() { alert('foo'); } // a function - Correct

func(); // call the function - alerts 'foo' - Correct

var func2 = function () { alert('foo'); } // same as 'func' surely? - Nope, func2 is a different object, that apparently does the same thing when called.

var Class = function() { alert('bar'); }; - It's a function with no name stored in variable Class.

var c = new Class(); - Calls function stored in Class supplying new empty object as this and returning that object. Functions called as new functionA() are expected to work as constructors and prepare a newly created object (this). In your case - constructor does nothing with the object and just alerts bar.

You also get classes in ES6 that look like this:

//class
class Cat {
    //constructor
    constructor() {
        this.name = 'Snowball';
    }

    //method
    meow() {
        console.log('Hello, nyah! My name is ' + this.name + ' nyah!~');
    }
};

There are no classes in javascript. But, there are ways to make a function to behave like a class in other languages.

A very good explanation is given here 3 way to define a class in js

Also, found a very good reference for OOP in Javascript

Object is the base type in JavaScript i.e. all the user defined data types inherits from Object in one way or another. So if you define a function or a class [note as of now JS doesn't support class construct, but its proposed in ECMAScript version 6], it will implicitly inherit from Object type.

Classes are really used for encapsulating logical functions and properties into one type / entity and you can 'new' it up using constructor syntax. So if you define a 'Customer' class, you can instantiate it multiple times and each instance / object can have different values. They can even share the values if you define class level value using prototype.

Since JS doesn't support class construct at the moment, functions can really act as individual method as well as container for other functions or types.

I hope with ECMAScript 6 we will have clear separation between these constructs similar to what we have in other languages like C#, Java etc.