Use:
Textbox1.Text = DateTime.Today.ToShortDateString()
That is equivalent to calling ToString("d") on an instance of a date. You can check out other standard format strings on the DateTime.ToString method MSDN page.
EDIT: the above will address the presentation side of things for the textbox, but your SQL column is probably a DateTime type which will store a default time if not given. That is why you're seeing the "12:00:00 AM" time portion. When you retrieve the data from SQL you should handle it in your ASP.NET code-behind to format the data as shown above.
Is there anything specific you're trying to achieve with just the date portion? On a related note, here's some food for thought: Working with Date and/or Time values in SQL Server: Don't Format, Don't Convert -- just use DATETIME.