Here is the C# custom control (AutoSuggestTextBox) code.

[DefaultProperty("Text")]
[ToolboxData("<{0}:AutoSuggestTextBox runat=server></{0}:AutoSuggestTextBox>")]
public class AutoSuggestTextBox : WebControl
{
protected override void RenderContents(HtmlTextWriter output)
output.WriteLine("<script type=\"text/javascript\">");
output.WriteLine("$().ready(function() {");
output.WriteLine(string.Format("$(\"#{0}TextControl\").autocomplete(\"{0}.ashx\");", this.ID));
output.WriteLine("});</script>");
output.WriteLine(string.Format("<input type=\"text\" id=\"{0}TextControl\" name=\"{0}TextControl\" />", this.ID));
}
}

In the implementation, I need to give the output from the database as string, so I implemented the code in a asp.net handler(ASHX) file. And in the ProcessRequest method, I wrote the following code.

if (!string.IsNullOrEmpty(context.Request.QueryString["q"])) // remember, 'q' is the query the autosuggest will pass
        {
            SqlDataReader reader;
            string query = string.Format("SELECT [CityName] FROM [City] WHERE [CityName] LIKE '%{0}'", context.Request.QueryString["q"].ToString());
            using (SqlConnection connection = new SqlConnection("Server=mydbserver; User Id=sa;Password=mypassword;Database=mylookups;"))
            {
                connection.Open();
                using (SqlCommand command = new SqlCommand(query, connection))
                {
                    reader = command.ExecuteReader(CommandBehavior.CloseConnection);
                    while (reader.Read())
                    {
                        context.Response.Write(reader["CityName"].ToString() + Environment.NewLine);
                    }
                    reader.Close();
                }
                connection.Close();
            }
        }

Note: I am not promising the above code is optimised. [:)]

And the ASPX Page you need to add reference to following JS Files and Style.

    <script type="text/javascript" language="javascript" src="jquery.js"></script>
    <script src="jquery.autocomplete.js" type="text/javascript"></script>
    <style type="text/css">
        .ac_odd {background:#dddddd;}
        .ac_results { padding: 0px; border: 1px solid black; background-color: white; overflow: hidden; z-index: 99999; }
        .ac_results ul { width: 100%; list-style-position: outside; list-style: none; padding: 0; margin: 0; }
        .ac_results li { margin: 0px; padding: 2px 5px; cursor: default; display: block; font: menu; font-size:12px; line-height: 16px; overflow: hidden; }
        .ac_loading { background:url(loading.jpg) right no-repeat; }
        .ac_over { background-color: #0A246A; color: white; }
    </style>

We have completed almost everything, then compile the project, drag the Custom control from toolbox to ASPX page, where you refering the JS Files and CSS. The ASHX File name should be the name of the control. Like if the control’s name is AutoSuggestCity, the ASHX Filename should be AutoSuggestCity.ashx. Please modify the control code if you want to change it.

After running you will get a Textbox, type something there, you will get the suggestions. Here is the one I got.

You can download JQuery from JQuery website. – www.jquery.com. And autosuggest plugin from AutoSuggest plugin page.