You can't use .ClientID in javascript. You need to give it the actual ID. Since you are using master pages the ID can get messed up.
Things I did:
Corrected javascript ids.
Added ClientIDMode="Static" to your button 'opener'.
Remove the runat tag from the 'dialog' div this isn't necessary.
<asp:content id="Content1" contentplaceholderid="head" runat="Server" xmlns:asp="#unknown">
<link rel="stylesheet" href="//code.jquery.com/ui/1.11.2/themes/smoothness/jquery-ui.css">
<script src="//code.jquery.com/jquery-1.10.2.js"></script>
<script src="//code.jquery.com/ui/1.11.2/jquery-ui.js"></script>
<link rel="stylesheet" href="/resources/demos/style.css">
<script>
$(function() {
$(
autoOpen: false,
show: {
effect: "blind",
duration: 1000
},
hide: {
effect: "explode",
duration: 1000
}
});
$(
$(
});
});
</script>
</link></link></asp:content>
<asp:content id="Content2" contentplaceholderid="ContentPlaceHolder1" runat="Server" xmlns:asp="#unknown">
<div id="dialog" title="Basic dialog">
<p>This is an animated dialog which is useful for displaying information. The dialog window can be moved, resized and closed with the
</div>
<asp:button id="opener" runat="server" text="Button" clientidmode="Static" />
</asp:content>