You can do some analytics at your end-
The XFBML and HTML5 versions of the button allow you to subscribe to the edge.create
event in the Facebook SDK for JavaScript through FB.Event.subscribe. This JavaScript event will fire any time a button is clicked.
edge.create is fired when someone clicks a like button on your page to like something.
edge.remove is fired when someone clicks a like button on your page to unlike something.
Example-
var page_like_or_unlike_callback = function(url, html_element) {
console.log("page_like_or_unlike_callback");
console.log(url);
console.log(html_element);
}
// In your onload handler
FB.Event.subscribe('edge.create', page_like_or_unlike_callback);
FB.Event.subscribe('edge.remove', page_like_or_unlike_callback);
And some other ways mentioned in the FAQ section of the documentation: Like Button