Updated below for ES6

March 2013 and ES5

This MDN document describes extending classes well:

https://developer.mozilla.org/en-US/docs/JavaScript/Introduction_to_Object-Oriented_JavaScript

In particular, here is now they handle it:

// define the Person Class
function Person() {}


Person.prototype.walk = function(){
alert ('I am walking!');
};
Person.prototype.sayHello = function(){
alert ('hello');
};


// define the Student class
function Student() {
// Call the parent constructor
Person.call(this);
}


// inherit Person
Student.prototype = Object.create(Person.prototype);


// correct the constructor pointer because it points to Person
Student.prototype.constructor = Student;


// replace the sayHello method
Student.prototype.sayHello = function(){
alert('hi, I am a student');
}


// add sayGoodBye method
Student.prototype.sayGoodBye = function(){
alert('goodBye');
}


var student1 = new Student();
student1.sayHello();
student1.walk();
student1.sayGoodBye();


// check inheritance
alert(student1 instanceof Person); // true
alert(student1 instanceof Student); // true

Note that Object.create() is unsupported in some older browsers, including IE8:

If you are in the position of needing to support these, the linked MDN document suggests using a polyfill, or the following approximation:

function createObject(proto) {
function ctor() { }
ctor.prototype = proto;
return new ctor();
}

Using this like Student.prototype = createObject(Person.prototype) is preferable to using new Person() in that it avoids calling the parent's constructor function when inheriting the prototype, and only calls the parent constructor when the inheritor's constructor is being called.

May 2017 and ES6

Thankfully, the JavaScript designers have heard our pleas for help and have adopted a more suitable way of approaching this issue.

MDN has another great example on ES6 class inheritance, but I'll show the exact same set of classes as above reproduced in ES6:

class Person {
sayHello() {
alert('hello');
}


walk() {
alert('I am walking!');
}
}


class Student extends Person {
sayGoodBye() {
alert('goodBye');
}


sayHello() {
alert('hi, I am a student');
}
}


var student1 = new Student();
student1.sayHello();
student1.walk();
student1.sayGoodBye();


// check inheritance
alert(student1 instanceof Person); // true
alert(student1 instanceof Student); // true

Clean and understandable, just like we all want. Keep in mind, that while ES6 is pretty common, it's not supported everywhere:

Try this:

Function.prototype.extends = function(parent) {
this.prototype = Object.create(parent.prototype);
};


Monkey.extends(Monster);
function Monkey() {
Monster.apply(this, arguments); // call super
}

Edit: I put a quick demo here http://jsbin.com/anekew/1/edit. Note that extends is a reserved word in JS and you may get warnings when linting your code, you can simply name it inherits, that's what I usually do.

With this helper in place and using an object props as only parameter, inheritance in JS becomes a bit simpler:

Function.prototype.inherits = function(parent) {
this.prototype = Object.create(parent.prototype);
};


function Monster(props) {
this.health = props.health || 100;
}


Monster.prototype = {
growl: function() {
return 'Grrrrr';
}
};


Monkey.inherits(Monster);
function Monkey() {
Monster.apply(this, arguments);
}


var monkey = new Monkey({ health: 200 });


console.log(monkey.health); //=> 200
console.log(monkey.growl()); //=> "Grrrr"

If you don't like the prototype approach, because it doesn't really behave in a nice OOP-way, you could try this:

var BaseClass = function()
{
this.some_var = "foobar";


/**
* @return string
*/
this.someMethod = function() {
return this.some_var;
}
};


var MyClass = new Class({ extends: BaseClass }, function()
{
/**
* @param string value
*/
this.__construct = function(value)
{
this.some_var = value;
}
})

Using lightweight library (2k minified): https://github.com/haroldiedema/joii

This is an extension (excuse the pun) of elclanrs' solution to include detail on instance methods, as well as taking an extensible approach to that aspect of the question; I fully acknowledge that this is put together thanks to David Flanagan's "JavaScript: The Definitive Guide" (partially adjusted for this context). Note that this is clearly more verbose than other solutions, but would probably benefit in the long-term.

First we use David's simple "extend" function, which copies properties to a specified object:

function extend(o,p) {
for (var prop in p) {
o[prop] = p[prop];
}
return o;
}

Then we implement his Subclass definition utility:

function defineSubclass(superclass,     // Constructor of our superclass
constructor,  // Constructor of our new subclass
methods,      // Instance methods
statics) {    // Class properties
// Set up the prototype object of the subclass
constructor.prototype = Object.create(superclass.prototype);
constructor.prototype.constructor = constructor;
if (methods) extend(constructor.prototype, methods);
if (statics) extend(constructor, statics);
return constructor;
}

For the last bit of preparation we enhance our Function prototype with David's new jiggery-pokery:

Function.prototype.extend = function(constructor, methods, statics) {
return defineSubclass(this, constructor, methods, statics);
};

After defining our Monster class, we do the following (which is re-usable for any new Classes we want to extend/inherit):

var Monkey = Monster.extend(
// constructor
function Monkey() {
this.bananaCount = 5;
Monster.apply(this, arguments);    // Superclass()
},
// methods added to prototype
{
eatBanana: function () {
this.bananaCount--;
this.health++;
this.growl();
}
}
);

For traditional extending you can simply write superclass as constructor function, and then apply this constructor for your inherited class.

     function AbstractClass() {
this.superclass_method = function(message) {
// do something
};
}


function Child() {
AbstractClass.apply(this);
// Now Child will have superclass_method()
}

Example on angularjs:

http://plnkr.co/edit/eFixlsgF3nJ1LeWUJKsd?p=preview

app.service('noisyThing',
['notify',function(notify){
this._constructor = function() {
this.scream = function(message) {
message = message + " by " + this.get_mouth();
notify(message);
console.log(message);
};


this.get_mouth = function(){
return 'abstract mouth';
}
}
}])
.service('cat',
['noisyThing', function(noisyThing){
noisyThing._constructor.apply(this)
this.meow = function() {
this.scream('meooooow');
}
this.get_mouth = function(){
return 'fluffy mouth';
}
}])
.service('bird',
['noisyThing', function(noisyThing){
noisyThing._constructor.apply(this)
this.twit = function() {
this.scream('fuuuuuuck');
}
}])

ES6 gives you now the opportunity to use class & extends keywords :

Then , your code will be :

You have a base class:

class Monster{
constructor(){
this.health = 100;
}
growl() {
console.log("Grr!");
}


}

That You want to extend and create another class with:

class Monkey extends Monster {
constructor(){
super(); //don't forget "super"
this.bananaCount = 5;
}




eatBanana() {
this.bananaCount--;
this.health++; //Accessing variable from parent class monster
this.growl(); //Accessing function from parent class monster
}


}

For Autodidacts:

function BaseClass(toBePrivate){
var morePrivates;
this.isNotPrivate = 'I know';
// add your stuff
}
var o = BaseClass.prototype;
// add your prototype stuff
o.stuff_is_never_private = 'whatever_except_getter_and_setter';




// MiddleClass extends BaseClass
function MiddleClass(toBePrivate){
BaseClass.call(this);
// add your stuff
var morePrivates;
this.isNotPrivate = 'I know';
}
var o = MiddleClass.prototype = Object.create(BaseClass.prototype);
MiddleClass.prototype.constructor = MiddleClass;
// add your prototype stuff
o.stuff_is_never_private = 'whatever_except_getter_and_setter';






// TopClass extends MiddleClass
function TopClass(toBePrivate){
MiddleClass.call(this);
// add your stuff
var morePrivates;
this.isNotPrivate = 'I know';
}
var o = TopClass.prototype = Object.create(MiddleClass.prototype);
TopClass.prototype.constructor = TopClass;
// add your prototype stuff
o.stuff_is_never_private = 'whatever_except_getter_and_setter';




// to be continued...

Create "instance" with getter and setter:

function doNotExtendMe(toBePrivate){
var morePrivates;
return {
// add getters, setters and any stuff you want
}
}

I can propose one variant, just have read in book, it seems the simplest:

function Parent() {
this.name = 'default name';
};


function Child() {
this.address = '11 street';
};


Child.prototype = new Parent();      // child class inherits from Parent
Child.prototype.constructor = Child; // constructor alignment


var a = new Child();


console.log(a.name);                // "default name" trying to reach property of inherited class

Summary:

There are multiple ways which can solve the problem of extending a constructor function with a prototype in Javascript. Which of these methods is the 'best' solution is opinion based. However, here are two frequently used methods in order to extend a constructor's function prototype.

ES 2015 Classes:

class Monster {
constructor(health) {
this.health = health
}
  

growl () {
console.log("Grr!");
}
  

}




class Monkey extends Monster {
constructor (health) {
super(health) // call super to execute the constructor function of Monster
this.bananaCount = 5;
}
}


const monkey = new Monkey(50);


console.log(typeof Monster);
console.log(monkey);

The above approach of using ES 2015 classes is nothing more than syntactic sugar over the prototypal inheritance pattern in javascript. Here the first log where we evaluate typeof Monster we can observe that this is function. This is because classes are just constructor functions under the hood. Nonetheless you may like this way of implementing prototypal inheritance and definitively should learn it. It is used in major frameworks such as ReactJS and Angular2+.

Factory function using Object.create():

function makeMonkey (bananaCount) {
  

// here we define the prototype
const Monster = {
health: 100,
growl: function() {
console.log("Grr!");}
}
  

const monkey = Object.create(Monster);
monkey.bananaCount = bananaCount;


return monkey;
}




const chimp = makeMonkey(30);


chimp.growl();
console.log(chimp.bananaCount);

This method uses the Object.create() method which takes an object which will be the prototype of the newly created object it returns. Therefore we first create the prototype object in this function and then call Object.create() which returns an empty object with the __proto__ property set to the Monster object. After this we can initialize all the properties of the object, in this example we assign the bananacount to the newly created object.

the absolutely minimal (and correct, unlike many of the answers above) version is:

function Monkey(param){
this.someProperty = param;
}
Monkey.prototype = Object.create(Monster.prototype);
Monkey.prototype.eatBanana = function(banana){ banana.eat() }

That's all. You can read here the longer explanation