Why would a JavaScript variable start with a dollar sign?

In the 1st, 2nd, and 3rd Edition of ECMAScript, using $-prefixed variable names was explicitly discouraged by the spec except in the context of autogenerated code:

The dollar sign ($) and the underscore (_) are permitted anywhere in an identifier. The dollar sign is intended for use only in mechanically generated code.

However, in the next version (the 5th Edition, which is current), this restriction was dropped, and the above passage replaced with

The dollar sign ($) and the underscore (_) are permitted anywhere in an IdentifierName.

As such, the $ sign may now be used freely in variable names. Certain frameworks and libraries have their own conventions on the meaning of the symbol, noted in other answers here.

As others have mentioned the dollar sign is intended to be used by mechanically generated code. However, that convention has been broken by some wildly popular JavaScript libraries. JQuery, Prototype and MS AJAX (AKA Atlas) all use this character in their identifiers (or as an entire identifier).

In short you can use the $ whenever you want. (The interpreter won't complain.) The question is when do you want to use it?

I personally do not use it, but I think its use is valid. I think MS AJAX uses it to signify that a function is an alias for some more verbose call.

For example:

var $get = function(id) { return document.getElementById(id); }

That seems like a reasonable convention.

Very common use in jQuery is to distinguish jQuery objects stored in variables from other variables.

For example, I would define:

var $email = $("#email"); // refers to the jQuery object representation of the dom object
var email_field = $("#email").get(0); // refers to the dom object itself

I find this to be very helpful in writing jQuery code and makes it easy to see jQuery objects which have a different set of properties.