Show textbox on change of dropdown and selected value is other

State dropdown have list of states with last item OTHER, so if user selects "OTHER" then a textbox should be displayed.

If user selects any state: 1. don't show text box 2. if already displayed, hide that

If user selects OTHER only then textbox should be displayed.

Adjust your HTML like this

<asp:DropDownList ID="ddlProducts" runat="server"  
     onchange="ShowHideOther(this, 'divOther');"/>
<div id="divOther" style="display:none">
   <asp:TextBox ID="txtOther" runat="server" />
</div>

<script>
function ShowHideOther(ddl, other)
{
   if(ddl.value == "OTHER")
     document.getElementById(other).style.display = "block";
   else
     document.getElementById(other).style.display = "none";     
}
</script>

try this by using jQyery:

$("#TextBoxID").hide();
$("#DropdownID").change(function(){
    if ($(this).val() === "YOURVALUE")
        $("#TextBoxID").show();
    else
        $("#TextBoxID").hide();
});