It's not a jQuery feature but a feature for debugging purposes. You can for instance log something to the console when something happens. For instance:
$('#someButton').click(function() {
console.log('#someButton was clicked');
// do something
});
You'd then see #someButton was clicked in Firebug’s “Console” tab (or another tool’s console — e.g. Chrome’s Web Inspector) when you would click the button.
For some reasons, the console object could be unavailable. Then you could check if it is - this is useful as you don't have to remove your debugging code when you deploy to production:
if (window.console && window.console.log) {
// console is available
}