Can I pass a JavaScript variable to another browser window?

Putting code to the matter, you can do this from the parent window:

var thisIsAnObject = {foo:'bar'};
var w = window.open("http://example.com");
w.myVariable = thisIsAnObject;

or this from the new window:

var myVariable = window.opener.thisIsAnObject;

I prefer the latter, because you will probably need to wait for the new page to load anyway, so that you can access its elements, or whatever you want.

Provided the windows are from the same security domain, and you have a reference to the other window, yes.

Javascript's open() method returns a reference to the window created (or existing window if it reuses an existing one). Each window created in such a way gets a property applied to it "window.opener" pointing to the window which created it.

Either can then use the DOM (security depending) to access properties of the other one, or its documents,frames etc.

Yes, scripts can access properties of other windows in the same domain that they have a handle on (typically gained through window.open/opener and window.frames/parent). It is usually more manageable to call functions defined on the other window rather than fiddle with variables directly.

However, windows can die or move on, and browsers deal with it differently when they do. Check that a window (a) is still open (!window.closed) and (b) has the function you expect available, before you try to call it.

Simple values like strings are fine, but generally it isn't a good idea to pass complex objects such as functions, DOM elements and closures between windows. If a child window stores an object from its opener, then the opener closes, that object can become 'dead' (in some browsers such as IE), or cause a memory leak. Weird errors can ensue.