How to insert data into SQL Server

I think you lack to pass Connection object to your command object. and it is much better if you will use command and parameters for that.

using (SqlConnection connection = new SqlConnection("ConnectionStringHere"))
{
    using (SqlCommand command = new SqlCommand())
    {
        command.Connection = connection;            // <== lacking
        command.CommandType = CommandType.Text;
        command.CommandText = "INSERT into tbl_staff (staffName, userID, idDepartment) VALUES (@staffName, @userID, @idDepart)";
        command.Parameters.AddWithValue("@staffName", name);
        command.Parameters.AddWithValue("@userID", userId);
        command.Parameters.AddWithValue("@idDepart", idDepart);

        try
        {
            connection.Open();
            int recordsAffected = command.ExecuteNonQuery();
        }
        catch(SqlException)
        {
            // error here
        }
        finally
        {
            connection.Close();
        }
    }
}

You have to set Connection property of Command object and use parametersized query instead of hardcoded SQL to avoid SQL Injection.

 using(SqlConnection openCon=new SqlConnection("your_connection_String"))
    {
      string saveStaff = "INSERT into tbl_staff (staffName,userID,idDepartment) VALUES (@staffName,@userID,@idDepartment)";

      using(SqlCommand querySaveStaff = new SqlCommand(saveStaff))
       {
         querySaveStaff.Connection=openCon;
         querySaveStaff.Parameters.Add("@staffName",SqlDbType.VarChar,30).Value=name;
         .....
         openCon.Open();

         querySaveStaff.ExecuteNonQuery();
       }
     }