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

There's a difference.

var x = 1 declares variable x in current scope (aka execution context). If the declaration appears in a function - a local variable is declared; if it's in global scope - a global variable is declared.

x = 1, on the other hand, is merely a property assignment. It first tries to resolve x against scope chain. If it finds it anywhere in that scope chain, it performs assignment; if it doesn't find x, only then does it creates x property on a global object (which is a top level object in a scope chain).

Now, notice that it doesn't declare a global variable, it creates a global property.

The difference between the two is subtle and might be confusing unless you understand that variable declarations also create properties (only on a Variable Object) and that every property in Javascript (well, ECMAScript) have certain flags that describe their properties - ReadOnly, DontEnum and DontDelete.

Since variable declaration creates property with the DontDelete flag, the difference between var x = 1 and x = 1 (when executed in global scope) is that the former one - variable declaration - creates the DontDelete'able property, and latter one doesn't. As a consequence, the property created via this implicit assignment can then be deleted from the global object, and the former one - the one created via variable declaration - cannot be deleted.

But this is just theory of course, and in practice there are even more differences between the two, due to various bugs in implementations (such as those from IE).

Hope it all makes sense : )

---

[Update 2010/12/16]

In ES5 (ECMAScript 5; recently standardized, 5th edition of the language) there's a so-called "strict mode" — an opt-in language mode, which slightly changes the behavior of undeclared assignments. In strict mode, assignment to an undeclared identifier is a ReferenceError. The rationale for this was to catch accidental assignments, preventing creation of undesired global properties. Some of the newer browsers have already started rolling support for strict mode. See, for example, my compat table.

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