First get all the files with specified extension using Directory.GetFiles() and then iterate through each files in the list and move them to target directory.
//Assume user types .txt into textbox
string fileExtension = "*" + textbox1.Text;
string[] txtFiles = Directory.GetFiles("Source Path", fileExtension);
foreach (var item in txtFiles)
{
File.Move(item, Path.Combine("Destination Directory", Path.GetFileName(item)));
}