Showing Results in DropDown list while typing in textbox asp.net C#

jQuery UI Autocomplete
This is called an auto-complete textbox. There is a jQuery UI widget available for this: http://jqueryui.com/autocomplete/.

In particular, click the "Remote with cache" example (from the examples list on the right of the above page) to see how to use a callback. You can click "view source" to see how they created their example. Note that you don't have to do the caching.

That callback will call a C# web service that you build. To call a web service from JavasScript, see this: http://msdn.microsoft.com/en-us/library/bb398998(v=vs.90).aspx.

You can start by creating a simple web service that just returns a list of strings (say, your 3 favorite animals, or whatever you want). Then, once you have that working, you can modify your web service to look at the term being searched for, then perform some query (e.g., against a database) to return valid results.

Pure ASP.NET Alternative
If you don't want to use jQuery, you can use this: http://www.asp.net/ajaxLibrary/AjaxControlToolkitSampleSite/AutoComplete/AutoComplete.aspx.