First of all I would advice not to save the department name in the employee table. You already have a relationship between the tables (Dept_ID) so you can always fetch the referenced department name by using a join. If you would save the department name to the employee, when the department name changes, you would need also to change the name in all corresponding employee rows.
Conserning the saving, to add to
Abhinav S's[
^] answer, the thing is that when you fetch the data into your user interface, you need to fetch both department name and id. This information could be stored inside a datatable, collection etc but the important thing is that when you bind the data to the drop down list, you need to have access to the id based on the text value the user selected.
Depending on the technology you use, some controls provide ability to bind display member (department name) and data member (department id) separately or you can bind the whole object (i.e. datarow etc) for a specific drop down item.
When saving you fetch the selected key value and use it in the statement, like in the example in solution 1, but without department name, so something like
string query = "INSERT INTO dbo.employee (dept_id,emp_name) VALUES (@deptid, @empname)";
command.Parameters.AddWithValue("@deptid",deptid_value)
command.Parameters.AddWithValue("@empname",empname_value)
command.ExecuteNonQuery();