Check the below conditions:
i). Set AutoPostBack=True
ii). Handle the eventhandler for the DropDownlist.
For Example:
Design:
<asp:dropdownlist runat="server" id="ddlTest" autopostback="True" onselectindexchanged="ddlTest_IndexChanged" xmlns:asp="#unknown" />
<asp:label runat="server" id="lblTest" xmlns:asp="#unknown" />
Code Behind:
protected void ddlTest_IndexChanged(Object sender, EventArgs e)
{
lblTest.Text=ddlTest.SelectedValue.ToString();
}