Details: Sounds like you don't have AutoPostBack set for the first drop down list. Example below
<asp:DropDownList ID="DropDownList1" runat="server" OnSelectedIndexChanged="DropDownList1_SelectedIndexChanged"
AutoPostBack="true">
<asp:ListItem>Cat</asp:ListItem>
<asp:ListItem>Dog</asp:ListItem>
</asp:DropDownList>
<asp:DropDownList ID="DropDownList2" runat="server"></asp:DropDownList>
Code Behind
protected void DropDownList1_SelectedIndexChanged(object sender, EventArgs e)
{
DropDownList2.Items.Add(DropDownList1.SelectedItem.Text);
}