Please refer this example
HTML
<div style="max-width: 200; float: left;">
    <asp:ListBox ID="lbLeftListBox" runat="server" SelectionMode="Multiple">
        <asp:ListItem Text="PC" Value="PC" />
        <asp:ListItem Text="LapTop HP" Value="LapTop HP" />
        <asp:ListItem Text="DELL" Value="DELL" />
        <asp:ListItem Text="Key Board" Value="Key Board" />
    </asp:ListBox>
</div>
<div style="max-width: 300; float: left; margin-left: 5px; margin-top: 10px;">
     
    <asp:LinkButton ID="lnkPassRight" Text=">>" OnClick="PassRight" runat="server"></asp:LinkButton>
    <br />
    <asp:LinkButton ID="lnkPassLeft" Text="<<" OnClick="PassLeft" runat="server"></asp:LinkButton>
     
</div>
<div style="width: 300; margin-left: 10px;">
    <asp:ListBox ID="lbRightListBox" runat="server" SelectionMode="Multiple"></asp:ListBox>
</div>
</form>
C#
protected void PassRight(object sender, EventArgs e)
{
    foreach (ListItem item in this.lbLeftListBox.Items)
    {
        if (item.Selected)
        {
            this.lbRightListBox.Items.Add(new ListItem(item.Text, item.Value));
        }
    }
    foreach (ListItem item in this.lbRightListBox.Items)
    {
        this.lbLeftListBox.Items.Remove(item);
    }
}
protected void PassLeft(object sender, EventArgs e)
{
    foreach (ListItem item in this.lbRightListBox.Items)
    {
        if (item.Selected)
        {
            this.lbLeftListBox.Items.Add(new ListItem(item.Text, item.Value));
        }
    }
    foreach (ListItem item in this.lbLeftListBox.Items)
    {
        this.lbRightListBox.Items.Remove(item);
    }
}