You should use the AutoComplete textbox from the AJAXControlToolkit. It has the exact functionality you are requesting, and the fetching of results is partially baked in by allowing you to wire up a service call to get the data.
Check out the following like which has an example of what you are needing. Start typing in some text and the results will begin to display in a listbox fasion below the textbox. Click on the 'AutoComplete Properties' link on the page to see how it works.
AutoComplete Demonstration:
http://www.asp.net/ajax/ajaxcontroltoolkit/Samples/AutoComplete/AutoComplete.aspx
Then check out this video for more help:
[How Do I:] Use the ASP.NET AJAX AutoComplete Control:
http://www.asp.net/ajax/videos/how-do-i-use-the-aspnet-ajax-autocomplete-control