The below code snippet explains how to get the Text and Index part of the Selected item of ASP.Net CheckBoxList using jQuery
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
    <title></title>
    <script type="text/javascript" src="http://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
    <script type="text/javascript">
        $("#demo").live("click", function () {
            var selectedItems = $("[id*=CheckBoxList1] input:checked");
            if (selectedItems.length > 0) {
                for (var i = 0; i < selectedItems.length; i++)
                    alert("Selected Text: " + selectedItems.eq(i).next().html() + " Selected Index: " + selectedItems[i].name.split("$")[selectedItems[i].name.split("$").length - 1]);
            } else {
                alert("Please select an item.");
            }
        });
    </script>
</head>
<body>
    <form id="form1" runat="server">
    <asp:CheckBoxList ID="CheckBoxList1" runat="server">
        <asp:ListItem Text="One" Value="1" />
        <asp:ListItem Text="Two" Value="2" />
        <asp:ListItem Text="Three" Value="3" />
    </asp:CheckBoxList>
    <br />
    <input type="button" id="demo" value="Demo" />
    </form>
</body>
</html>
 
 
Explanation:
In the above code snippet I have assigned a click event handler to the HTML input button with ID demo. When the button is clicked the selected item (radio button) of the ASP.Net CheckBoxList is fetched using the [id*= CheckBoxList1] input:checked  jQuery selector. If the length of the selected item is greater than zero the Index and the Text Part of the selected item of the ASP.Net CheckBoxList is displayed in JavaScript alert.
Note: ASP.Net RadioButtonList stores the value part in ViewState hence it is not possible to fetch the checked item value of ASP.Net CheckBoxList.
Demo: