Search files in a folder

In the Visual Studio designer, highlight your TextBox.
Look at the Properties pane.
Click the button that looks like a lightning bolt - that will show you the Events instead of the Properties.
Scroll down to TextChanged. Double click it.
That will hook up an event handler for you, and take you to the method that is called when the user changes the text in the text box.

You can now write your code in the method - when the user types in the TextBox, that method will be called for each character.

The first thing to do is add this to the using section if not there already:

using System.IO;
using System.Data;

To get the files from a folder, you can use:

string[] foundFiles = Directory.GetFiles(thePathVariable, theSearchBoxText + "*.*");

You have to change "thePathVariable" and "theSearchBoxText" to suite your needs.
By appending the wildcard string ("*.*") you're searching for every file with any extension that starts with the text written in your search box. You can play with the wildcards to achieve any other needs.

To display the results into a datagridview, we have to transfer the data from the string array to a datatable. This can be also achieved easily with LINQ, but seems you're not quite there, so:

DataTable s = new DataTable();
s.Columns.Add("FileName");

foreach (string aFile in foundFiles)
{
   s.Rows.Add(aFile);
}
dataGridView1.DataSource = s;

And that's it, hope it helps.