TargetControlID will take ClientID of the link button. If AnimationExtender and LinkButton is there in the same panel or control then ClientID of LinkButton will be easily accessible by AnimationExtender. Try this:
<asp:gridview id="GridView1" runat="server" autogeneratecolumns="False" xmlns:asp="#unknown">
OnRowCommand="GridView1_RowCommand" onrowediting="GridView1_RowEditing" >
<columns>
<asp:boundfield headertext="ID" datafield="ID" />
<asp:boundfield headertext="Name" datafield="ContactName" />
<asp:boundfield headertext="Company" datafield="ContCompany" />
<asp:boundfield headertext="Profession" datafield="ContProfession" />
<asp:boundfield headertext="Address1" datafield="ContAdd1" />
<asp:boundfield headertext="Address2" datafield="ContAdd2" />
<asp:boundfield headertext="City" datafield="ContCity" />
<asp:boundfield headertext="Pincode" datafield="ContPincode" />
<asp:boundfield headertext="Type" datafield="ContType" />
<asp:boundfield headertext="Status" datafield="ContStatus" />
<asp:templatefield>
<itemtemplate>
<asp:linkbutton id="Edit" runat="server" onclick="Show_Button_click" text="Edit" causesvalidation="False" />
<asp:animationextender id="popupanimation" runat="server" targetcontrolid="Edit">
<animations>
<onclick>
<parallel animationtarget="Panel1">
Duration="0.3" Fps="25">
<fadein />
</parallel>
</onclick>
</animations>
</asp:animationextender>
<asp:panel id="Panel1" runat="server"></asp:panel>
</itemtemplate>
</asp:templatefield>
</columns>
</asp:gridview>
--Amit