What is the purpose of the var keyword and when should I use it (or omit it)?

NOTE: This question was asked from the viewpoint of ECMAScript version 3 or 5. The answers might become outdated with the introduction of new features in the release of ECMAScript 6.

What exactly is the function of the var keyword in JavaScript, and what is the difference between

var someNumber = 2;
var someFunction = function() { doSomething; }
var someObject = { }
var someObject.someProperty = 5;

and

someNumber = 2;
someFunction = function() { doSomething; }
someObject = { }
someObject.someProperty = 5;

?

When would you use either one, and why/what does it do?

If you're in the global scope then there's not much difference. Read Kangax's answer for explanation

If you're in a function then var will create a local variable, "no var" will look up the scope chain until it finds the variable or hits the global scope (at which point it will create it):

// These are both globals
var foo = 1;
bar = 2;

function()
{
    var foo = 1; // Local
    bar = 2;     // Global

    // Execute an anonymous function
    (function()
    {
        var wibble = 1; // Local
        foo = 2; // Inherits from scope above (creating a closure)
        moo = 3; // Global
    }())
}

If you're not doing an assignment then you need to use var:

var x; // Declare x

Stack Overflow Q#1470488, score: 1413